BackPrecalculus Test 1 Review – Step-by-Step Study Guidance
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Q1. Factor
Background
Topic: Polynomial Factoring
This question tests your ability to factor a cubic polynomial by grouping and recognizing common factors.
Key Terms and Formulas:
Factoring by grouping: Rearranging and grouping terms to factor common elements.
Polynomial: An expression consisting of variables and coefficients.
Step-by-Step Guidance
Group the terms: and .
Factor out the greatest common factor (GCF) from each group.
Look for a common binomial factor in both groups.
Try solving on your own before revealing the answer!
Q2. Solve for : \frac{x+6}{3} = \frac{x+8}{8}$
Background
Topic: Solving Rational Equations
This question tests your ability to solve equations involving fractions by finding a common denominator or cross-multiplying.
Key Terms and Formulas:
Rational equation: An equation involving fractions with variables in the numerator and/or denominator.
Step-by-Step Guidance
Cross-multiply to eliminate the denominators: .
Expand both sides of the equation.
Collect like terms and isolate .
Try solving on your own before revealing the answer!
Q3. Solve for :
Background
Topic: Quadratic Equations
This question tests your ability to solve quadratic equations by rearranging and factoring or using the quadratic formula.
Key Terms and Formulas:
Quadratic equation:
Quadratic formula:
Step-by-Step Guidance
Move all terms to one side: .
Identify , , and for the quadratic formula.
Set up the quadratic formula with these values.
Try solving on your own before revealing the answer!
Q4. Solve:
Background
Topic: Solving Quadratic Equations
This question tests your ability to isolate in a quadratic equation and solve for its value.
Key Terms and Formulas:
Quadratic equation:
Step-by-Step Guidance
Subtract 3 from both sides to isolate the quadratic term.
Divide both sides by 8 to solve for .
Take the square root of both sides, remembering to consider both positive and negative roots.
Try solving on your own before revealing the answer!
Q5. Solve the absolute value inequality and express the solution set in interval notation:
Background
Topic: Absolute Value Inequalities
This question tests your understanding of how to solve inequalities involving absolute values and how to express the solution in interval notation.
Key Terms and Formulas:
Absolute value: is the distance from to 0 on the number line.
To solve , rewrite as .
Step-by-Step Guidance
Add 2 to both sides to isolate the absolute value: .
Rewrite as a compound inequality: .
Solve for by adding 4 to all parts of the inequality.
Try solving on your own before revealing the answer!
Q6. Use the graph below to determine the function’s domain and range
Background
Topic: Domain and Range from Graphs
This question tests your ability to interpret a graph and identify the set of possible input values (domain) and output values (range) for a function.
Key Terms and Formulas:
Domain: All possible -values for which the function is defined.
Range: All possible -values the function can take.
Step-by-Step Guidance
Examine the graph and identify the leftmost and rightmost points to determine the domain.
Identify the lowest and highest points on the graph to determine the range.
Express your answers in interval notation.
Try solving on your own before revealing the answer!
Q7. Identify the intervals that the function above is increasing
Background
Topic: Increasing and Decreasing Intervals
This question tests your ability to analyze a graph and determine where the function is increasing (where increases as increases).
Key Terms and Formulas:
Increasing interval: Where the function rises as you move from left to right.
Step-by-Step Guidance
Look for sections of the graph where the curve moves upward as you move right.
Identify the -intervals corresponding to these sections.
Express the intervals using interval notation.
Try solving on your own before revealing the answer!
Q8. Evaluate the piecewise function at the given value:
Background
Topic: Piecewise Functions
This question tests your ability to evaluate a function defined by different expressions depending on the value of .
Key Terms and Formulas:
Piecewise function: A function defined by different expressions for different intervals of the domain.
Step-by-Step Guidance
For , use the rule for .
For , use the rule for and substitute into the expression.
Simplify the expressions as needed.
Try solving on your own before revealing the answer!
Q9. Find and simplify the difference quotient , for
Background
Topic: Difference Quotient
This question tests your ability to compute and simplify the difference quotient, which is foundational for understanding derivatives in calculus.
Key Terms and Formulas:
Difference quotient:
Step-by-Step Guidance
Find by substituting into .
Set up the difference quotient: .
Combine the fractions in the numerator over a common denominator.
Simplify the expression as much as possible.
Try solving on your own before revealing the answer!
Q10. Write the equation for the line passing through and in point-slope form and in slope-intercept form
Background
Topic: Equations of Lines
This question tests your ability to find the equation of a line given two points, using both point-slope and slope-intercept forms.
Key Terms and Formulas:
Slope formula:
Point-slope form:
Slope-intercept form:
Step-by-Step Guidance
Calculate the slope using the two points.
Write the equation in point-slope form using one of the points.
Rearrange to slope-intercept form by solving for .
Try solving on your own before revealing the answer!
Q11. Graph the line
Background
Topic: Graphing Linear Equations
This question tests your ability to graph a line given in slope-intercept form.
Key Terms and Formulas:
Slope-intercept form:
Slope (): Rise over run
Y-intercept (): Where the line crosses the -axis
Step-by-Step Guidance
Plot the -intercept at .
Use the slope to find another point: up 1 unit, right 2 units.
Draw a straight line through the points.
Try solving on your own before revealing the answer!
Q12. Graph the function
Background
Topic: Graphing Linear Equations
This question tests your ability to rearrange a linear equation and graph it.
Key Terms and Formulas:
Standard form:
Slope-intercept form:
Step-by-Step Guidance
Rearrange the equation to solve for in terms of .
Identify the slope and -intercept.
Plot the -intercept and use the slope to find another point.
Draw the line through the points.
Try solving on your own before revealing the answer!
Q13. Given the functions and , determine:
a.
Background
Topic: Function Operations
This question tests your ability to add two functions together.
Key Terms and Formulas:
Sum of functions:
Step-by-Step Guidance
Write explicitly.
Combine like terms if possible.
Try solving on your own before revealing the answer!
b.
Background
Topic: Function Operations
This question tests your ability to subtract one function from another.
Key Terms and Formulas:
Difference of functions:
Step-by-Step Guidance
Write explicitly.
Combine like terms if possible.
Try solving on your own before revealing the answer!
c.
Background
Topic: Function Multiplication
This question tests your ability to multiply two functions together.
Key Terms and Formulas:
Product of functions:
Step-by-Step Guidance
Multiply and together.
Expand and simplify the resulting expression.
Try solving on your own before revealing the answer!
d.
Background
Topic: Function Division
This question tests your ability to divide one function by another and state any restrictions on the domain.
Key Terms and Formulas:
Quotient of functions:
Step-by-Step Guidance
Write explicitly.
State any restrictions where .
Try solving on your own before revealing the answer!
Q14. Given and , find and state the domain
Background
Topic: Function Composition
This question tests your ability to compose two functions and determine the domain of the composite function.
Key Terms and Formulas:
Composition:
Step-by-Step Guidance
Substitute into wherever you see .
Simplify the resulting expression.
State the domain based on any restrictions from or .
Try solving on your own before revealing the answer!
Q15. Find functions and so that ;
Background
Topic: Function Decomposition
This question tests your ability to express a function as a composition of two simpler functions.
Key Terms and Formulas:
Composition:
Step-by-Step Guidance
Identify an inner function that could be substituted into an outer function to produce .
Write in terms of and identify and .
Try solving on your own before revealing the answer!
Q16. Determine which two functions are inverses of each other if given: , ,
Background
Topic: Inverse Functions
This question tests your ability to identify pairs of functions that are inverses by composing them and checking if the result is .
Key Terms and Formulas:
Inverse functions: and
Step-by-Step Guidance
Compose each pair of functions and simplify to see if the result is .
Check both and for each pair.
Try solving on your own before revealing the answer!
Q17. Find the inverse of the one-to-one function:
Background
Topic: Finding Inverse Functions
This question tests your ability to find the inverse of a linear function by solving for in terms of and then switching variables.
Key Terms and Formulas:
Inverse function:
Step-by-Step Guidance
Replace with .
Solve for in terms of .
Switch and to write the inverse function.
Try solving on your own before revealing the answer!
