{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1. Find the equation for the circle with a diameter whose endpoints are (-4, -2) and (2, 6)."}]},{"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: Equations of Circles"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of how to write the equation of a circle given the endpoints of its diameter."}]},{"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":"Center of a circle: midpoint of the diameter"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Radius: half the distance between the endpoints of the diameter"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Standard equation: "},{"type":"inlineMath","attrs":{"latex":" (x - h)^2 + (y - k)^2 = r^2 "}}]}]}]},{"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 midpoint of the diameter using "},{"type":"inlineMath","attrs":{"latex":" \\left( \\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2} \\right) "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate the distance between the endpoints using "},{"type":"inlineMath","attrs":{"latex":" \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide the distance by 2 to get the radius."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the equation using the center and radius found above."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The center is at "},{"type":"inlineMath","attrs":{"latex":"(-1, 2)"}},{"type":"text","text":" and the radius is $5"},{"type":"inlineMath","attrs":{"latex":". The equation is "}},{"type":"text","text":" (x + 1)^2 + (y - 2)^2 = 25 $."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"We used the midpoint formula for the center and the distance formula for the radius."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2. Find the domain of the function "},{"type":"inlineMath","attrs":{"latex":" f(x) = \\frac{1 - 4x}{2x - 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: Domain of Rational Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to determine the domain of a function, specifically where the denominator is not zero."}]},{"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: set of all real numbers for which the function is defined"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rational function: function of the form "},{"type":"inlineMath","attrs":{"latex":" \\frac{P(x)}{Q(x)} "}}]}]}]},{"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 denominator: "},{"type":"inlineMath","attrs":{"latex":"2x - 1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set the denominator equal to zero: "},{"type":"inlineMath","attrs":{"latex":"2x - 1 = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to find the value that makes the denominator zero."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Exclude this value from the domain."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The domain is all real numbers except "},{"type":"inlineMath","attrs":{"latex":" x = \\frac{1}{2} "}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The function is undefined when the denominator is zero."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3. Find the equation of the line passing through "},{"type":"inlineMath","attrs":{"latex":"(-3, 2)"}},{"type":"text","marks":[{"type":"bold"}],"text":" that is perpendicular to the line "},{"type":"inlineMath","attrs":{"latex":"3x + 4y = 5"}},{"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: Equations of Lines and Perpendicularity"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the equation of a line given a point and a perpendicular relationship."}]},{"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":"Slope-intercept form: "},{"type":"inlineMath","attrs":{"latex":" y = mx + b "}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Perpendicular slopes: "},{"type":"inlineMath","attrs":{"latex":" m_1 \\cdot m_2 = -1 "}}]}]}]},{"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 "},{"type":"inlineMath","attrs":{"latex":"3x + 4y = 5"}},{"type":"text","text":" in slope-intercept form to find its slope."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the negative reciprocal of the slope for the perpendicular line."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the point "},{"type":"inlineMath","attrs":{"latex":"(-3, 2)"}},{"type":"text","text":" and the perpendicular slope in the point-slope form: "},{"type":"inlineMath","attrs":{"latex":" y - y_1 = m(x - x_1) "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify to get the equation in slope-intercept or standard form."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The equation is "},{"type":"inlineMath","attrs":{"latex":" y = -\\frac{3}{4}x - \\frac{1}{4} "}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"We found the slope of the given line, took its negative reciprocal, and used the point-slope formula."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4. Given "},{"type":"inlineMath","attrs":{"latex":" f(x) = 2x + 3 "}},{"type":"text","marks":[{"type":"bold"}],"text":", find and simplify (if possible) the difference quotient "},{"type":"inlineMath","attrs":{"latex":" \\frac{f(x + h) - f(x)}{h} "}},{"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: Difference Quotient"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to compute and simplify the difference quotient, which is foundational for calculus."}]},{"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":"Difference quotient: "},{"type":"inlineMath","attrs":{"latex":" \\frac{f(x + h) - f(x)}{h} "}}]}]}]},{"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":" f(x + h) "}},{"type":"text","text":" by substituting "},{"type":"inlineMath","attrs":{"latex":" x + h "}},{"type":"text","text":" into "},{"type":"inlineMath","attrs":{"latex":" f(x) "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Subtract "},{"type":"inlineMath","attrs":{"latex":" f(x) "}},{"type":"text","text":" from "},{"type":"inlineMath","attrs":{"latex":" f(x + h) "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide the result by "},{"type":"inlineMath","attrs":{"latex":" h "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the expression as much as possible."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The simplified difference quotient is $2$."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For a linear function, the difference quotient yields the slope."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5. Given "},{"type":"inlineMath","attrs":{"latex":" f(x) = 3x + 1 "}},{"type":"text","marks":[{"type":"bold"}],"text":" and "},{"type":"inlineMath","attrs":{"latex":" g(x) = 2x^2 - 6x "}},{"type":"text","marks":[{"type":"bold"}],"text":", find "},{"type":"inlineMath","attrs":{"latex":" (f \\circ g)(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: Function Composition"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to compose two functions and evaluate at a specific value."}]},{"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":"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":"Evaluate "},{"type":"inlineMath","attrs":{"latex":" g(2) "}},{"type":"text","text":" by substituting "},{"type":"inlineMath","attrs":{"latex":" x = 2 "}},{"type":"text","text":" into "},{"type":"inlineMath","attrs":{"latex":" g(x) "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the result from "},{"type":"inlineMath","attrs":{"latex":" g(2) "}},{"type":"text","text":" as the input for "},{"type":"inlineMath","attrs":{"latex":" f(x) "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Evaluate "},{"type":"inlineMath","attrs":{"latex":" f(g(2)) "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the expression."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":" (f \\circ g)(2) = -11 "}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"We substituted "},{"type":"inlineMath","attrs":{"latex":" x = 2 "}},{"type":"text","text":" into "},{"type":"inlineMath","attrs":{"latex":" g(x) "}},{"type":"text","text":", then used that value in "},{"type":"inlineMath","attrs":{"latex":" f(x) "}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6. Describe how the graph of "},{"type":"inlineMath","attrs":{"latex":" f(x) = (x + 2)^2 - 1 "}},{"type":"text","marks":[{"type":"bold"}],"text":" can be obtained from the graph of "},{"type":"inlineMath","attrs":{"latex":" y = 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: Transformations of Quadratic Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of how to apply shifts to the basic quadratic graph."}]},{"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 shift: "},{"type":"inlineMath","attrs":{"latex":" x \\rightarrow x + h "}},{"type":"text","text":" shifts left by "},{"type":"inlineMath","attrs":{"latex":" h "}},{"type":"text","text":" units"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Vertical shift: "},{"type":"inlineMath","attrs":{"latex":" y \\rightarrow y + k "}},{"type":"text","text":" shifts up by "},{"type":"inlineMath","attrs":{"latex":" k "}},{"type":"text","text":" units"}]}]}]},{"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 horizontal shift: "},{"type":"inlineMath","attrs":{"latex":" x + 2 "}},{"type":"text","text":" means shift left by 2 units."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the vertical shift: "},{"type":"inlineMath","attrs":{"latex":" -1 "}},{"type":"text","text":" means shift down by 1 unit."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Describe the sequence of transformations applied to "},{"type":"inlineMath","attrs":{"latex":" y = x^2 "}},{"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":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Shift the graph of "},{"type":"inlineMath","attrs":{"latex":" y = x^2 "}},{"type":"text","text":" left by 2 units and down by 1 unit."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The transformations are horizontal and vertical shifts."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q7. Given that "},{"type":"inlineMath","attrs":{"latex":" f(x) = \\frac{4x + 3}{x - 4} "}},{"type":"text","marks":[{"type":"bold"}],"text":" is a one-to-one function, find a formula for the inverse function."}]},{"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":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"To find the inverse: swap "},{"type":"inlineMath","attrs":{"latex":" x "}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":" y "}},{"type":"text","text":" and solve for $ y $"}]}]}]},{"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":"Let "},{"type":"inlineMath","attrs":{"latex":" y = \\frac{4x + 3}{x - 4} "}},{"type":"text","text":"."}]}]},{"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 get "},{"type":"inlineMath","attrs":{"latex":" x = \\frac{4y + 3}{y - 4} "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"type":"inlineMath","attrs":{"latex":" y "}},{"type":"text","text":" in terms of "},{"type":"inlineMath","attrs":{"latex":" x "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the expression to get "},{"type":"inlineMath","attrs":{"latex":" f^{-1}(x) "}},{"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":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":" f^{-1}(x) = \\frac{4x - 3}{x - 4} "}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"We swapped variables and solved for "},{"type":"inlineMath","attrs":{"latex":" y "}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8. Find the zeros of the function "},{"type":"inlineMath","attrs":{"latex":" f(x) = 2x^2 + 3x + 8 "}},{"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: Quadratic Equations and Zeros"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the roots of a quadratic 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":"Quadratic formula: "},{"type":"inlineMath","attrs":{"latex":" x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a} "}}]}]}]},{"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 "},{"type":"inlineMath","attrs":{"latex":" a = 2 "}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":" b = 3 "}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":" c = 8 "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Plug these values into the quadratic formula."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate the discriminant "},{"type":"inlineMath","attrs":{"latex":" b^2 - 4ac "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the expression for "},{"type":"inlineMath","attrs":{"latex":" x "}},{"type":"text","text":" using the formula."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The zeros are "},{"type":"inlineMath","attrs":{"latex":" x = \\frac{-3 \\pm i \\sqrt{55}}{4} "}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The discriminant is negative, so the zeros are complex."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q9. 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Determine the end behavior of "},{"type":"inlineMath","attrs":{"latex":" f(x) = (x - 1)^3 + 3x^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: End Behavior of Polynomials"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to analyze the end behavior of a polynomial 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":"End behavior: how "},{"type":"inlineMath","attrs":{"latex":" f(x) "}},{"type":"text","text":" behaves as "},{"type":"inlineMath","attrs":{"latex":" x \\rightarrow \\infty "}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":" x \\rightarrow -\\infty "}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Degree and leading coefficient determine end behavior"}]}]}]},{"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":"Expand "},{"type":"inlineMath","attrs":{"latex":" (x - 1)^3 "}},{"type":"text","text":" to find the degree and leading term."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Add "},{"type":"inlineMath","attrs":{"latex":" 3x^2 "}},{"type":"text","text":" to the expanded expression."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the highest degree term and its coefficient."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Describe the end behavior based on the degree and leading coefficient."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"As "},{"type":"inlineMath","attrs":{"latex":" x \\rightarrow \\infty "}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":" f(x) \\rightarrow \\infty "}},{"type":"text","text":"; as "},{"type":"inlineMath","attrs":{"latex":" x \\rightarrow -\\infty "}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":" f(x) \\rightarrow -\\infty "}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The cubic term dominates the end behavior."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q14. 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Which of the following functions has an inverse function?"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":" f(x) = x^3 - 2 "}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":" g(x) = x^2 + 6 "}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":" h(x) = x^2 - 2 "}}]}]}]},{"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 and One-to-One Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to determine which functions are one-to-one and thus have inverses."}]},{"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":"One-to-one function: passes the horizontal line test"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Inverse exists only for one-to-one functions"}]}]}]},{"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 each function to see if it is one-to-one."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that cubic functions are one-to-one, but quadratic functions are not."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the horizontal line test to each 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":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":" f(x) = x^3 - 2 "}},{"type":"text","text":" has an inverse function."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"It is one-to-one, unlike the quadratic functions."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q16. Approximate "},{"type":"inlineMath","attrs":{"latex":" \\log_{0.3}(100) "}},{"type":"text","marks":[{"type":"bold"}],"text":" to 4 decimal places."}]},{"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 Change of Base"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to evaluate logarithms with non-standard bases."}]},{"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":"Change of base formula: "},{"type":"inlineMath","attrs":{"latex":" \\log_b(a) = \\frac{\\log(a)}{\\log(b)} "}}]}]}]},{"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 change of base formula: "},{"type":"inlineMath","attrs":{"latex":" \\log_{0.3}(100) = \\frac{\\log(100)}{\\log(0.3)} "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate "},{"type":"inlineMath","attrs":{"latex":" \\log(100) "}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":" \\log(0.3) "}},{"type":"text","text":" using a calculator."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide the results to get the value."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Round to 4 decimal places."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":" \\log_{0.3}(100) \\approx -8.3842 "}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"We used the change of base formula and rounded to 4 decimal places."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q17. A wood sculptor carves three types of statues: Totem Poles, Bears, and Deers. Each statue requires carving, sanding, and painting. If the sculptor spends a total of 14 hours carving, 15 hours sanding, and 21 hours painting, how many of each type of statue were made? Use Gauss elimination or Gauss-Jordan elimination with matrices."}]},{"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: Systems of Linear Equations and Matrices"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to set up and solve a system of equations using matrix methods."}]},{"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":"System of equations: three variables"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Matrix representation and row operations"}]}]}]},{"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 up the system:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":" 2T + 2B + 1D = 14 "}},{"type":"text","text":" (carving)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":" 1T + 2B + 2D = 15 "}},{"type":"text","text":" (sanding)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":" 3T + 2B + 2D = 21 "}},{"type":"text","text":" (painting)"}]}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the augmented matrix for the system."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply Gauss or Gauss-Jordan elimination to reduce the matrix."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"type":"inlineMath","attrs":{"latex":" T "}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":" B "}},{"type":"text","text":", and "},{"type":"inlineMath","attrs":{"latex":" D "}},{"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":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Totem Poles: "},{"type":"inlineMath","attrs":{"latex":" 3 "}},{"type":"text","text":"; Bears: "},{"type":"inlineMath","attrs":{"latex":" 4 "}},{"type":"text","text":"; Deers: "},{"type":"inlineMath","attrs":{"latex":" 2 "}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"We solved the system using matrix row operations."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q18. Expand "},{"type":"inlineMath","attrs":{"latex":" \\log_3 \\left( \\frac{4x^3}{8y^2} \\right) "}},{"type":"text","marks":[{"type":"bold"}],"text":" as much as possible."}]},{"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: Logarithm Properties"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to expand logarithmic expressions using properties."}]},{"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":" \\log_b \\left( \\frac{A}{B} \\right) = \\log_b(A) - \\log_b(B) "}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":" \\log_b(A^n) = n \\log_b(A) "}}]}]}]},{"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 quotient rule: "},{"type":"inlineMath","attrs":{"latex":" \\log_3(4x^3) - \\log_3(8y^2) "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Expand each logarithm using the product and power rules."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express "},{"type":"inlineMath","attrs":{"latex":" 4 "}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":" 8 "}},{"type":"text","text":" as powers of "},{"type":"inlineMath","attrs":{"latex":" 2 "}},{"type":"text","text":" if possible."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine like terms and simplify."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":" \\log_3(4) + 3\\log_3(x) - \\log_3(8) - 2\\log_3(y) "}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"We used logarithm properties to expand the expression."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q19. Solve "},{"type":"inlineMath","attrs":{"latex":" \\log_2(20) + \\log_2(3x - 2) = \\log_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: Logarithmic Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve equations involving logarithms."}]},{"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":" \\log_b(A) + \\log_b(B) = \\log_b(AB) "}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set arguments equal if logs are equal"}]}]}]},{"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 the left side using the product rule: "},{"type":"inlineMath","attrs":{"latex":" \\log_2(20(3x - 2)) "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set "},{"type":"inlineMath","attrs":{"latex":" 20(3x - 2) = x + 4 "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve the resulting linear equation for "},{"type":"inlineMath","attrs":{"latex":" x "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check for extraneous solutions."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":" x = \\frac{44}{59} "}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"We combined logs and solved the resulting equation."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q20. Solve "},{"type":"inlineMath","attrs":{"latex":" 5e^{2x} = 40 "}},{"type":"text","marks":[{"type":"bold"}],"text":". Approximate your answer to 4 decimal places."}]},{"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 exponential equations using logarithms."}]},{"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":"Take the natural logarithm of both sides"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":" e^{a} = b \\implies a = \\ln(b) "}}]}]}]},{"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":"Divide both sides by "},{"type":"inlineMath","attrs":{"latex":" 5 "}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":" e^{2x} = 8 "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Take the natural logarithm: "},{"type":"inlineMath","attrs":{"latex":" 2x = \\ln(8) "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"type":"inlineMath","attrs":{"latex":" x "}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":" x = \\frac{\\ln(8)}{2} "}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use a calculator to approximate "},{"type":"inlineMath","attrs":{"latex":" x "}},{"type":"text","text":" to 4 decimal places."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":" x \\approx 1.0397 "}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"We used logarithms to solve for "},{"type":"inlineMath","attrs":{"latex":" x "}},{"type":"text","text":" and rounded to 4 decimal places."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q21. In 2011 the population of Texas was 25.67 million with a growth rate of 2.06% per year. Estimate the population size in 2022 using the growth model "},{"type":"inlineMath","attrs":{"latex":" P(t) = P_0 e^{kt} "}},{"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 Growth Models"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use exponential models to estimate population growth."}]},{"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":" P(t) = P_0 e^{kt} "}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":" k "}},{"type":"text","text":" is the growth rate (as a decimal)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":" t "