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Quizzes > Mathematics > Advanced Math

Master Pre-Calculus: Functions & Equations Practice Test

Dive into our advanced pre-calculus final exam quiz for functions, equations, and analytical thinking challenges!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art scene with graphs symbols equations and pen on sky blue background promoting free pre calculus practice test

This pre-calculus practice test helps you practice functions and equations so you can spot weak areas fast. Work through short problems, see what sticks, and use the quick chapter review to brush up on rules before an exam; then try again to track progress.

Given f(x) = 2x + 3, what is f(4)?
10
8
9
11
To evaluate f(4), substitute x = 4 into f(x) = 2x + 3, yielding 2 * 4 + 3 = 11. Function evaluation involves replacing the variable with the given input value. This process is fundamental to understanding how functions map inputs to outputs. For more examples and practice, see Functions Evaluation.
What is the slope and y-intercept of the linear function y = -3x + 5?
Slope is 3; y-intercept is 5
Slope is -3; y-intercept is 5
Slope is -3; y-intercept is -5
Slope is 5; y-intercept is -3
In slope-intercept form y = mx + b, m represents the slope and b represents the y-intercept. In the equation y = -3x + 5, the slope m is -3 and the y-intercept b is 5. Identifying these parameters helps graph linear functions quickly. Learn more at .
What is the domain of the function f(x) = ?(x - 1)?
x > 1
x ? 1
x ? 1
All real x
A square root function f(x) = ?(x - 1) requires the radicand x - 1 to be non-negative, so x - 1 ? 0 and thus x ? 1. The domain of f is all real x that satisfy this inequality. Understanding domain restrictions is critical for functions involving radicals. See Domain of Radical Functions for more detail.
If f(x) = 2x + 1 and g(x) = x², what is (g ? f)(3)?
49
16
64
14
The composition (g ? f)(3) means g(f(3)). First, f(3) = 2 * 3 + 1 = 7, then g(7) = 7² = 49. Composition applies functions sequentially, passing the output of the inner function as the input to the outer one. For more examples, visit Function Composition.
What is the inverse function f?¹(x) of f(x) = (x - 3)/2?
(x - 3)/2
2x + 3
2x - 3
(x + 3)/2
To find the inverse, swap x and y in y = (x - 3)/2, giving x = (y - 3)/2, then solve for y: multiply both sides by 2 to get 2x = y - 3 and add 3: y = 2x + 3. Therefore, f?¹(x) = 2x + 3. Inverse functions reverse the input-output roles of the original function. Additional practice can be found at Inverse Functions.
Solve the quadratic equation x² - 5x + 6 = 0.
x = -2 or x = -3
x = 2 or x = 3
x = 1 or x = 6
x = 5 or x = -1
The quadratic factors as (x - 2)(x - 3) = 0, so the solutions are x = 2 and x = 3. Factoring is a common method for solving second-degree polynomials when integer roots exist. Verifying solutions by substitution confirms both satisfy the original equation. See for more information.
Solve the equation |2x - 3| = 7.
x = -5 or x = -2
x = 5 or x = -2
x = -5 or x = 2
x = 2 or x = -5
We solve |2x - 3| = 7 by considering two cases: 2x - 3 = 7, which gives x = 5; and 2x - 3 = -7, which gives x = -2. Absolute value equations always yield two linear equations unless the right side is zero. Checking both solutions ensures they satisfy the original equation. See for more examples.
What is the horizontal asymptote of the function f(x) = (3x² - x + 2)/(x² + 4)?
y = -1
y = 0
y = 3
y = 1
For a rational function where the degrees of numerator and denominator are equal, the horizontal asymptote is y = ratio of leading coefficients. Here both polynomials have degree 2, and the leading coefficients are 3 and 1, so y = 3/1 = 3. Horizontal asymptotes describe end-behavior of functions as x approaches ±?. More details at .
Solve the equation log?(x) + log?(x - 2) = 2.
4
9
1 - ?10
1 + ?10
Using log properties, combine logs: log?[x(x - 2)] = 2, so x(x - 2) = 3² = 9 and x² - 2x - 9 = 0. Solving gives x = [2 ± ?(4 + 36)]/2 = 1 ± ?10. We reject 1 - ?10 since it is negative and outside the domain of the original logs, leaving x = 1 + ?10. See for further study.
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Study Outcomes

  1. Analyze Function Behaviors -

    Develop the ability to examine and classify different types of functions, including linear, quadratic, and exponential, within a pre-calculus practice test framework.

  2. Solve Equations Efficiently -

    Master techniques for solving a variety of equations - such as polynomial, rational, and radical - through targeted practice in the functions and equations quiz.

  3. Interpret Graphical Data -

    Learn to read and interpret function graphs accurately, identifying key features like intercepts, asymptotes, and intervals of increase or decrease.

  4. Apply Function Transformations -

    Use transformations, compositions, and inverses to manipulate functions and understand their effects on graphs and equations.

  5. Evaluate Analytical Thinking -

    Enhance problem-solving strategies by working through advanced pre-calculus questions that test logical reasoning and analytical skills.

  6. Track Progress and Identify Gaps -

    Monitor your performance throughout the pre-calculus final exam quiz to pinpoint areas for review and reinforce your understanding.

Cheat Sheet

  1. Master Function Notation and Domain/Range -

    Understanding f(x) notation is essential for any pre-calculus practice test, as it lays the foundation for evaluating and interpreting functions. For instance, f(x)=√(x−2) requires x≥2 to keep the expression real, giving you a quick way to check valid inputs and outputs (Stewart, Calculus). Regularly consult MIT OpenCourseWare for clear definitions and examples to build confidence.

  2. Visualize Function Transformations -

    Recognizing how y=f(x−h)+k shifts and stretches graphs can save time on your functions and equations quiz. A simple mnemonic - "Horizontal before Vertical" (H then V) - helps remember to apply x-shifts before y-shifts, such as y=(x−2)²+3 moving right 2 and up 3 (OpenStax Precalculus). Practice sketching these in batches to see patterns emerge quickly.

  3. Solve Quadratic and Polynomial Equations Efficiently -

    Employ the quadratic formula x=[−b±√(b²−4ac)]/(2a) when factoring gets tricky, or use the Rational Root Theorem for higher-order polynomials (Khan Academy). For example, 2x³−3x²−8x+12=0 can be tackled by testing divisors of 12, leading to faster factorization. Refine these techniques through past pre-calculus final exam quiz problems to boost speed and accuracy.

  4. Master Composition and Inverse Functions -

    Function composition (f∘g)(x)=f(g(x)) and finding inverses f❻¹(x) "undo" each operation in reverse order - think "PEMDAS backwards" as a handy trick (Paul's Online Math Notes). For example, if f(x)=2x+5, solving y=2x+5 for x gives f❻¹(x)=(x−5)/2. Solidify this by swapping x and y then isolating y until it becomes natural.

  5. Apply Analytical Thinking to Systems of Equations -

    Whether using substitution, elimination, or matrix methods, flexible approaches to 2×2 or 3×3 systems sharpen your analytical thinking math test skills. For example, solve {2x+y=7, 3x−y=4} by adding equations to eliminate y, yielding x=11/5, then back-substitute for y. Draw on National Council of Teachers of Mathematics (NCTM) problem sets for real-world contexts and deeper understanding.

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