BackPrecalculus Test 4 Study Guide: Inequalities, Functions, Logarithms, and Exponentials
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1(a). Solve the inequality: "},{"type":"inlineMath","attrs":{"latex":"x + 12 < 7"}},{"type":"text","marks":[{"type":"bold"}],"text":" and express your answer in interval notation."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Linear Inequalities"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve a basic linear inequality and express the solution using interval notation."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Linear inequality: An inequality involving a linear expression."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Interval notation: A way to describe sets of numbers between endpoints."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Start by isolating "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" on one side of the inequality. Subtract $12"},{"type":"inlineMath","attrs":{"latex":" from both sides to get "}},{"type":"text","text":"x$ alone."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the resulting inequality and simplify the right side."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express the solution set in interval notation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1(b). Solve the inequality: "},{"type":"inlineMath","attrs":{"latex":"2x^3 > -8x^2"}},{"type":"text","marks":[{"type":"bold"}],"text":" and express your answer in interval notation."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Polynomial Inequalities"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve inequalities involving polynomials by factoring and analyzing sign changes."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Polynomial inequality: An inequality involving a polynomial expression."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factoring: Expressing a polynomial as a product of its factors."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Test intervals: Checking the sign of the expression in intervals determined by the roots."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Move all terms to one side to set the inequality to $0"},{"type":"inlineMath","attrs":{"latex":": $2x^3 + 8x^2 > 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factor the expression completely."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set each factor equal to zero to find critical points."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Draw a sign chart or test values in each interval to determine where the expression is positive."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1(c). Solve the inequality: "},{"type":"inlineMath","attrs":{"latex":"\\frac{x-1}{x+2} \\geq -2"}},{"type":"text","marks":[{"type":"bold"}],"text":" and express your answer in interval notation."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Rational Inequalities"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve inequalities involving rational expressions and to express the solution in interval notation."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rational inequality: An inequality involving a ratio of polynomials."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Critical points: Values where the numerator or denominator is zero."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine all terms on one side to get zero on the other: "},{"type":"inlineMath","attrs":{"latex":"\\frac{x-1}{x+2} + 2 \\geq 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find a common denominator and combine the terms into a single rational expression."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the numerator and set the numerator and denominator equal to zero to find critical points."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Test intervals between the critical points to determine where the expression is non-negative."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1(d). Solve the inequality: "},{"type":"inlineMath","attrs":{"latex":"\\frac{x-1}{x} \\geq 0"}},{"type":"text","marks":[{"type":"bold"}],"text":" and express your answer in interval notation."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Rational Inequalities"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve rational inequalities and express the solution in interval notation."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rational expression: A fraction where the numerator and denominator are polynomials."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Non-negativity: The expression is greater than or equal to zero."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the values where the numerator and denominator are zero: "},{"type":"inlineMath","attrs":{"latex":"x-1=0"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"x=0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Determine the sign of the expression in each interval defined by these points."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Include points where the expression equals zero, but exclude points where the denominator is zero."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2. Find the domain of "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\ln\\left(\\frac{x-1}{x+1}\\right)"}},{"type":"text","marks":[{"type":"bold"}],"text":", expressing your answer in interval notation."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Domain of Logarithmic Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the domain restrictions for logarithmic functions, especially when the argument is a rational expression."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Domain: The set of all real numbers for which the function is defined."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Logarithm: "},{"type":"inlineMath","attrs":{"latex":"\\ln(x)"}},{"type":"text","text":" is defined only for "},{"type":"inlineMath","attrs":{"latex":"x > 0"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set the argument of the logarithm greater than zero: "},{"type":"inlineMath","attrs":{"latex":"\\frac{x-1}{x+1} > 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve the inequality by finding where the numerator and denominator change sign."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express the solution in interval notation, excluding points where the denominator is zero."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2(a). For "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\frac{x}{x+3}"}},{"type":"text","marks":[{"type":"bold"}],"text":" and "},{"type":"inlineMath","attrs":{"latex":"g(x) = \\frac{2}{x}"}},{"type":"text","marks":[{"type":"bold"}],"text":", find the domains of "},{"type":"inlineMath","attrs":{"latex":"f \\circ g"}},{"type":"text","marks":[{"type":"bold"}],"text":" and "},{"type":"inlineMath","attrs":{"latex":"g \\circ f"}},{"type":"text","marks":[{"type":"bold"}],"text":". Then calculate "},{"type":"inlineMath","attrs":{"latex":"(f \\circ g)(x)"}},{"type":"text","marks":[{"type":"bold"}],"text":" and "},{"type":"inlineMath","attrs":{"latex":"(g \\circ f)(x)"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Composition of Functions and Domains"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the domain of composite functions and to compute the composition explicitly."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Composition: "},{"type":"inlineMath","attrs":{"latex":"(f \\circ g)(x) = f(g(x))"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Domain: The set of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" values for which the composition is defined."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the domain of "},{"type":"inlineMath","attrs":{"latex":"g(x)"}},{"type":"text","text":": Identify values of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" that make the denominator zero."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the domain of "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":": Identify values of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" that make the denominator zero."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For "},{"type":"inlineMath","attrs":{"latex":"f \\circ g"}},{"type":"text","text":", determine for which "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" both "},{"type":"inlineMath","attrs":{"latex":"g(x)"}},{"type":"text","text":" is defined and "},{"type":"inlineMath","attrs":{"latex":"f(g(x))"}},{"type":"text","text":" is defined."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For "},{"type":"inlineMath","attrs":{"latex":"g \\circ f"}},{"type":"text","text":", determine for which "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" both "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" is defined and "},{"type":"inlineMath","attrs":{"latex":"g(f(x))"}},{"type":"text","text":" is defined."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compute "},{"type":"inlineMath","attrs":{"latex":"(f \\circ g)(x)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"(g \\circ f)(x)"}},{"type":"text","text":" by substituting "},{"type":"inlineMath","attrs":{"latex":"g(x)"}},{"type":"text","text":" into "},{"type":"inlineMath","attrs":{"latex":"f"}},{"type":"text","text":" and vice versa."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2(b). For "},{"type":"inlineMath","attrs":{"latex":"f(x) = 3x^2 - 7"}},{"type":"text","marks":[{"type":"bold"}],"text":", "},{"type":"inlineMath","attrs":{"latex":"g(x) = 2x - a"}},{"type":"text","marks":[{"type":"bold"}],"text":", find "},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","marks":[{"type":"bold"}],"text":" such that the graph of "},{"type":"inlineMath","attrs":{"latex":"f \\circ g"}},{"type":"text","marks":[{"type":"bold"}],"text":" crosses the "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","marks":[{"type":"bold"}],"text":"-axis at $68$."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Function Composition and Intercepts"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of function composition and how to find parameters so that the graph passes through a specific point."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Composition: "},{"type":"inlineMath","attrs":{"latex":"(f \\circ g)(x) = f(g(x))"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-intercept: The value of the function when "},{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the composition "},{"type":"inlineMath","attrs":{"latex":"f(g(x))"}},{"type":"text","text":" explicitly."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set "},{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","text":" to find the "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-intercept of "},{"type":"inlineMath","attrs":{"latex":"f \\circ g"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set "},{"type":"inlineMath","attrs":{"latex":"f(g(0)) = 68"}},{"type":"text","text":" and solve for "},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2(c). For "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\frac{x+1}{x}"}},{"type":"text","marks":[{"type":"bold"}],"text":" and "},{"type":"inlineMath","attrs":{"latex":"g(x) = 1 - x"}},{"type":"text","marks":[{"type":"bold"}],"text":", compute "},{"type":"inlineMath","attrs":{"latex":"(f \\circ g)(0)"}},{"type":"text","marks":[{"type":"bold"}],"text":" and "},{"type":"inlineMath","attrs":{"latex":"(g \\circ f)(1)"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Function Composition and Evaluation"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to evaluate composite functions at specific values."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Composition: "},{"type":"inlineMath","attrs":{"latex":"(f \\circ g)(x) = f(g(x))"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Evaluation: Substitute the given value into the composite function."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compute "},{"type":"inlineMath","attrs":{"latex":"g(0)"}},{"type":"text","text":", then substitute this value into "},{"type":"inlineMath","attrs":{"latex":"f"}},{"type":"text","text":" to find "},{"type":"inlineMath","attrs":{"latex":"(f \\circ g)(0)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compute "},{"type":"inlineMath","attrs":{"latex":"f(1)"}},{"type":"text","text":", then substitute this value into "},{"type":"inlineMath","attrs":{"latex":"g"}},{"type":"text","text":" to find "},{"type":"inlineMath","attrs":{"latex":"(g \\circ f)(1)"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3(a). For "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\frac{3}{x+2}"}},{"type":"text","marks":[{"type":"bold"}],"text":", state the domain of "},{"type":"inlineMath","attrs":{"latex":"f"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Domain of Rational Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the domain of a rational function."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Domain: All real numbers except where the denominator is zero."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set the denominator "},{"type":"inlineMath","attrs":{"latex":"x+2"}},{"type":"text","text":" not equal to zero and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express the domain in set notation or interval notation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3(b). For "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\frac{3}{x+2}"}},{"type":"text","marks":[{"type":"bold"}],"text":", state the range of "},{"type":"inlineMath","attrs":{"latex":"f"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Range of Rational Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to determine the range of a rational function."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Range: All possible output values of the function."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set "},{"type":"inlineMath","attrs":{"latex":"y = \\frac{3}{x+2}"}},{"type":"text","text":" and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" in terms of "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Determine if there are any restrictions on "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" based on the algebraic manipulation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3(c). Find the inverse function "},{"type":"inlineMath","attrs":{"latex":"f^{-1}(x)"}},{"type":"text","marks":[{"type":"bold"}],"text":" for "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\frac{3}{x+2}"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Inverse Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the inverse of a rational function."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Inverse function: "},{"type":"inlineMath","attrs":{"latex":"f^{-1}(x)"}},{"type":"text","text":" satisfies "},{"type":"inlineMath","attrs":{"latex":"f(f^{-1}(x)) = x"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Replace "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" with "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" in terms of $y$."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Swap "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" to write the inverse function."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3(d). State the domain of "},{"type":"inlineMath","attrs":{"latex":"f^{-1}"}},{"type":"text","marks":[{"type":"bold"}],"text":" for "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\frac{3}{x+2}"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Domain of Inverse Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of how the range of the original function becomes the domain of the inverse."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Domain of "},{"type":"inlineMath","attrs":{"latex":"f^{-1}"}},{"type":"text","text":": All "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" for which "},{"type":"inlineMath","attrs":{"latex":"f^{-1}(x)"}},{"type":"text","text":" is defined."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall the range of "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" and use it as the domain for "},{"type":"inlineMath","attrs":{"latex":"f^{-1}(x)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify any values that would make the denominator of "},{"type":"inlineMath","attrs":{"latex":"f^{-1}(x)"}},{"type":"text","text":" zero."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4(a). For "},{"type":"inlineMath","attrs":{"latex":"f(x) = 2^{-x} - 4"}},{"type":"text","marks":[{"type":"bold"}],"text":", state the domain of "},{"type":"inlineMath","attrs":{"latex":"f"}},{"type":"text","marks":[{"type":"bold"}],"text":" in interval notation."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Domain of Exponential Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the domain of exponential functions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Exponential function: Defined for all real numbers unless otherwise restricted."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Consider the base and exponent to determine if there are any restrictions on "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express the domain in interval notation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4(b). For "},{"type":"inlineMath","attrs":{"latex":"f(x) = 2^{-x} - 4"}},{"type":"text","marks":[{"type":"bold"}],"text":", state the range of "},{"type":"inlineMath","attrs":{"latex":"f"}},{"type":"text","marks":[{"type":"bold"}],"text":" in interval notation."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Range of Exponential Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to determine the range of a transformed exponential function."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Range: All possible output values of the function."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Analyze the behavior of "},{"type":"inlineMath","attrs":{"latex":"2^{-x}"}},{"type":"text","text":" as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" approaches "},{"type":"inlineMath","attrs":{"latex":"\\infty"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"-\\infty"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Subtract $4$ to see how the range is shifted."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4(c). Find the "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","marks":[{"type":"bold"}],"text":"-intercept and "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","marks":[{"type":"bold"}],"text":"-intercept of "},{"type":"inlineMath","attrs":{"latex":"f(x) = 2^{-x} - 4"}},{"type":"text","marks":[{"type":"bold"}],"text":". Give the actual points."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Intercepts of Exponential Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find where the graph crosses the axes."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"-intercept: Set "},{"type":"inlineMath","attrs":{"latex":"f(x) = 0"}},{"type":"text","text":" and solve for $x$."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-intercept: Evaluate "},{"type":"inlineMath","attrs":{"latex":"f(0)"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set "},{"type":"inlineMath","attrs":{"latex":"2^{-x} - 4 = 0"}},{"type":"text","text":" and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to find the $x$-intercept."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Evaluate "},{"type":"inlineMath","attrs":{"latex":"f(0)"}},{"type":"text","text":" to find the "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-intercept."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4(d). Find the horizontal asymptote for the graph of "},{"type":"inlineMath","attrs":{"latex":"f(x) = 2^{-x} - 4"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Asymptotes of Exponential Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of horizontal asymptotes for exponential functions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Horizontal asymptote: The value that "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" approaches as "},{"type":"inlineMath","attrs":{"latex":"x \\to \\infty"}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":"x \\to -\\infty"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Analyze the end behavior of "},{"type":"inlineMath","attrs":{"latex":"2^{-x}"}},{"type":"text","text":" as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" approaches "},{"type":"inlineMath","attrs":{"latex":"\\infty"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"-\\infty"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Subtract $4$ to determine the horizontal asymptote."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4(e). Find "},{"type":"inlineMath","attrs":{"latex":"f^{-1}(x)"}},{"type":"text","marks":[{"type":"bold"}],"text":" for "},{"type":"inlineMath","attrs":{"latex":"f(x) = 2^{-x} - 4"}},{"type":"text","marks":[{"type":"bold"}],"text":". State the domain of "},{"type":"inlineMath","attrs":{"latex":"f^{-1}"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Inverse of Exponential Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the inverse of an exponential function and determine its domain."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Inverse function: Swap "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" and solve for $y$."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Domain of "},{"type":"inlineMath","attrs":{"latex":"f^{-1}"}},{"type":"text","text":": The range of "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" becomes the domain of "},{"type":"inlineMath","attrs":{"latex":"f^{-1}(x)"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Replace "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" with "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":", swap "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" and $y$, and solve for $y$."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the domain of "},{"type":"inlineMath","attrs":{"latex":"f^{-1}"}},{"type":"text","text":" based on the range of "},{"type":"inlineMath","attrs":{"latex":"f"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5(a). Find the exact value of "},{"type":"inlineMath","attrs":{"latex":"\\log_2(32)"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Logarithms and Exponents"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the relationship between logarithms and exponents."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Logarithm: "},{"type":"inlineMath","attrs":{"latex":"\\log_b(a) = x"}},{"type":"text","text":" means "},{"type":"inlineMath","attrs":{"latex":"b^x = a"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express $32"},{"type":"inlineMath","attrs":{"latex":" as a power of $2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the definition of logarithms to find the value."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5(b). Solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","marks":[{"type":"bold"}],"text":": "},{"type":"inlineMath","attrs":{"latex":"\\log_3(5x - 16) = 2"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Solving Logarithmic Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve equations involving logarithms by rewriting them in exponential form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Logarithmic equation: "},{"type":"inlineMath","attrs":{"latex":"\\log_b(a) = c"}},{"type":"text","text":" implies "},{"type":"inlineMath","attrs":{"latex":"a = b^c"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite the equation in exponential form."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" algebraically."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5(c). Express "},{"type":"inlineMath","attrs":{"latex":"\\frac{2}{3} \\ln 8 - \\ln(5^2 - 1)"}},{"type":"text","marks":[{"type":"bold"}],"text":" as a single logarithm in simplest form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Properties of Logarithms"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use properties of logarithms to combine and simplify expressions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Product rule: "},{"type":"inlineMath","attrs":{"latex":"\\ln a + \\ln b = \\ln(ab)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quotient rule: "},{"type":"inlineMath","attrs":{"latex":"\\ln a - \\ln b = \\ln\\left(\\frac{a}{b}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Power rule: "},{"type":"inlineMath","attrs":{"latex":"k \\ln a = \\ln(a^k)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the power rule to "},{"type":"inlineMath","attrs":{"latex":"\\frac{2}{3} \\ln 8"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify "},{"type":"inlineMath","attrs":{"latex":"5^2 - 1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine the terms using the quotient rule."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6(a). For "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\ln(x + 3)"}},{"type":"text","marks":[{"type":"bold"}],"text":", state the domain of "},{"type":"inlineMath","attrs":{"latex":"f"}},{"type":"text","marks":[{"type":"bold"}],"text":" in interval notation."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Domain of Logarithmic Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the domain of a logarithmic function."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Domain: The argument of "},{"type":"inlineMath","attrs":{"latex":"\\ln"}},{"type":"text","text":" must be positive."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set "},{"type":"inlineMath","attrs":{"latex":"x + 3 > 0"}},{"type":"text","text":" and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express the solution in interval notation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6(b). For "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\ln(x + 3)"}},{"type":"text","marks":[{"type":"bold"}],"text":", state the range of "},{"type":"inlineMath","attrs":{"latex":"f"}},{"type":"text","marks":[{"type":"bold"}],"text":" in interval notation."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Range of Logarithmic Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to determine the range of a logarithmic function."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Range: All possible output values of the function."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that the natural logarithm function has a range of all real numbers."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check if the transformation affects the range."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6(c). Find the "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","marks":[{"type":"bold"}],"text":"-intercept and "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","marks":[{"type":"bold"}],"text":"-intercept of "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\ln(x + 3)"}},{"type":"text","marks":[{"type":"bold"}],"text":". Give the actual points."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Intercepts of Logarithmic Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find where the graph crosses the axes."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"-intercept: Set "},{"type":"inlineMath","attrs":{"latex":"f(x) = 0"}},{"type":"text","text":" and solve for $x$."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-intercept: Evaluate "},{"type":"inlineMath","attrs":{"latex":"f(0)"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set "},{"type":"inlineMath","attrs":{"latex":"\\ln(x + 3) = 0"}},{"type":"text","text":" and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to find the $x$-intercept."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Evaluate "},{"type":"inlineMath","attrs":{"latex":"f(0)"}},{"type":"text","text":" to find the "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-intercept."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6(d). Find the vertical asymptote for the graph of "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\ln(x + 3)"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Asymptotes of Logarithmic Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of vertical asymptotes for logarithmic functions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Vertical asymptote: The value of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" where the argument of "},{"type":"inlineMath","attrs":{"latex":"\\ln"}},{"type":"text","text":" approaches zero from the right."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set "},{"type":"inlineMath","attrs":{"latex":"x + 3 = 0"}},{"type":"text","text":" and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to find the vertical asymptote."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6(e). Find "},{"type":"inlineMath","attrs":{"latex":"f^{-1}(x)"}},{"type":"text","marks":[{"type":"bold"}],"text":" for "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\ln(x + 3)"}},{"type":"text","marks":[{"type":"bold"}],"text":". State the domain of "},{"type":"inlineMath","attrs":{"latex":"f^{-1}"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Inverse of Logarithmic Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the inverse of a logarithmic function and determine its domain."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Inverse function: Swap "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" and solve for $y$."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Domain of "},{"type":"inlineMath","attrs":{"latex":"f^{-1}"}},{"type":"text","text":": The range of "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" becomes the domain of "},{"type":"inlineMath","attrs":{"latex":"f^{-1}(x)"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Replace "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" with "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":", swap "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" and $y$, and solve for $y$."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the domain of "},{"type":"inlineMath","attrs":{"latex":"f^{-1}"}},{"type":"text","text":" based on the range of "},{"type":"inlineMath","attrs":{"latex":"f"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q7(a). Solve the equation: "},{"type":"inlineMath","attrs":{"latex":"25^x - 8 \\cdot 5^x + 16 = 0"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Exponential Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve equations involving exponents by substitution or factoring."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Exponential equation: An equation where the variable is in the exponent."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitution: Let "},{"type":"inlineMath","attrs":{"latex":"y = 5^x"}},{"type":"text","text":" to simplify the equation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express "},{"type":"inlineMath","attrs":{"latex":"25^x"}},{"type":"text","text":" as "},{"type":"inlineMath","attrs":{"latex":"(5^2)^x = (5^x)^2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Let "},{"type":"inlineMath","attrs":{"latex":"y = 5^x"}},{"type":"text","text":" and rewrite the equation as a quadratic in "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve the quadratic equation for "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Back-substitute "},{"type":"inlineMath","attrs":{"latex":"y = 5^x"}},{"type":"text","text":" and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" using logarithms."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q7(b). Solve the equation: "},{"type":"inlineMath","attrs":{"latex":"\\ln(x + 6) - \\ln(x - 4) = \\ln x"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Logarithmic Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use properties of logarithms to combine and solve equations."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph