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Quizzes > Mathematics > Advanced Math

Precalculus Pre-Test Quiz: Prove You Can Ace It

Ready for precalculus practice problems? Tackle pre calc questions now!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art illustration for precalculus quiz with functions and equations on coral background

This free precalculus quiz helps you practice pre calc questions on functions, equations, and graphs. Take a short, scored set to spot gaps before a test. For a preview, see sample questions and answers or try another practice test now.

Given f(x) = 2x + 3, what is f(5)?
10
11
8
13
To evaluate a function, substitute the input value into its formula. Here, replacing x with 5 gives f(5)=2·5+3=10+3=13. This approach is fundamental to understanding how functions work. Learn more about evaluating functions.
What is the domain of the function f(x) = ?(x - 4)?
x ? 4
x < 4
x > 4
x ? 4
A square root requires its radicand to be nonnegative. Thus x - 4 must be ? 0, which gives x ? 4. Values below 4 would make the expression imaginary. .
Simplify x² · x³.
x³
x?
x?
x?
When multiplying with the same base, add exponents: x² · x³ = x^(2+3) = x?. This property holds for any real exponent values. .
Factor the polynomial x² - 5x + 6.
(x - 3)(x + 2)
(x - 1)(x - 6)
(x - 2)(x - 3)
(x + 2)(x + 3)
We look for two numbers that multiply to +6 and add to -5, which are -2 and -3. Thus x² - 5x + 6 factors to (x - 2)(x - 3). Factoring simplifies solving quadratic equations. .
Simplify 2³ · 2².
8
64
16
32
Add the exponents when multiplying like bases: 2³ · 2² = 2^(3+2) = 2? = 32. This is a basic exponent rule that applies universally. .
What is log??(1000)?
0
10
1000
3
Logarithms answer the question: 10 raised to what power equals 1000? Since 10³ = 1000, log??(1000) = 3. This is a fundamental definition of common logarithms. .
Simplify sin²? + cos²?.
0
1
sin2?
cos2?
The Pythagorean identity states sin²? + cos²? = 1 for all ?. This identity is foundational in trigonometry. .
What is the slope of the line given by y = 4x - 7?
4
-7
¼
-4
In the slope-intercept form y = mx + b, the coefficient m is the slope. Here, m = 4, so the line rises 4 units vertically for each unit it moves to the right. .
If f(x) = x + 1 and g(x) = 2x, find (f ? g)(3).
8
7
5
9
Composition (f ? g)(3) means f(g(3)). First compute g(3) = 2·3 = 6, then f(6) = 6 + 1 = 7. Function composition evaluates inner functions before outer. .
Solve the equation x² + 6x + 5 = 0.
x = 1 or x = -5
x = -1 or x = 5
x = 1 or x = 5
x = -1 or x = -5
This quadratic factors as (x + 1)(x + 5) = 0, giving roots x = -1 and x = -5. Factoring is a quick method for solving simple quadratics. .
Simplify the expression (x² - 9)/(x - 3).
3
x - 3
x² - 3
x + 3
Factor the numerator: x² - 9 = (x - 3)(x + 3). Cancel the (x - 3) terms to get x + 3, provided x ? 3. .
Find f?¹(7) if f(x) = 3x - 2.
1.5
7/3
5/3
3
To find the inverse, swap x and y: x = 3y - 2, then 3y = x + 2, so y = (x + 2)/3. Plugging x = 7 gives (7 + 2)/3 = 3. Inverse function guide.
Solve e? = 5 for x.
e?
5
ln(1/5)
ln(5)
Taking the natural log of both sides gives x = ln(5). Exponential and logarithm are inverse operations. .
What is log?(16)?
4
2
16
8
Since 2? = 16, log?(16) = 4. Logarithms convert exponent operations into multiplications. .
Find the amplitude and period of y = 3 sin(2x).
Amplitude = 2 and period = ?
Amplitude = 3 and period = ?
Amplitude = 3 and period = 2?
Amplitude = 1 and period = 2?
The coefficient 3 is the amplitude. Period = 2? divided by the frequency 2, so 2?/2 = ?. Trigonometric transformations alter these values. .
What is the domain of f(x) = log(x - 1)?
x ? 1
x > 0
x ? 0
x > 1
Inside a logarithm must be positive: x - 1 > 0, so x > 1. Log domain restrictions prevent invalid inputs. .
Given f(x) = x² and g(x) = ?x, what is (g ? f)(3)?
9
27
6
3
(g ? f)(3) = g(f(3)) = g(9) = ?9 = 3. Composition applies f first, then g. Composition of functions.
Solve the equation x/(x - 1) = 2.
2
0
1
-2
Multiply both sides by x - 1: x = 2(x - 1) = 2x - 2, then x - 2x = -2, so -x = -2 and x = 2. Exclude x = 1 to avoid division by zero. .
Solve x² = 2x + 3.
x = 1 or x = -3
x = -3 or x = -1
x = 3 or x = -1
x = 2 or x = -2
Rearrange: x² - 2x - 3 = 0, which factors as (x - 3)(x + 1) = 0. The solutions are x = 3 and x = -1. Quadratic formula or factoring both yield the same roots. .
Simplify tan(x) · cos(x).
tan²(x)
sin(x)
cos²(x)
1
tan(x) = sin(x)/cos(x), so tan(x)·cos(x) = sin(x). This leverages fundamental trig definitions. .
Determine the solution set for the inequality x² - 4x + 3 > 0.
x < 1 only
x > 3 only
x < 1 or x > 3
1 < x < 3
Factor the left side: (x - 1)(x - 3) > 0. The product is positive when both factors share sign: either x < 1 or x > 3. Graphical or sign-chart methods confirm this. .
What is the horizontal asymptote of y = (2x² + 3)/(x² - 1)?
y = 0
y = 3
x = 2
y = 2
For rational functions with equal-degree numerator and denominator, the horizontal asymptote is the ratio of leading coefficients: 2/1 = 2. This applies as x ? ±?. .
Find the inverse function of f(x) = (2x - 1)/(x + 3).
(x + 3)/(2x - 1)
(3x - 1)/(x + 2)
(2x - 1)/(x + 3)
-(3x + 1)/(x - 2)
Start with y = (2x - 1)/(x + 3). Swap x and y, then solve for y: x = (2y - 1)/(y + 3). Cross-multiply and isolate y to get y = -(3x + 1)/(x - 2). Inverse functions.
Divide x³ + 2x² - x + 5 by x + 1.
Quotient x² + x - 2 and remainder -7
Quotient x² + 2x - 1 and remainder 5
Quotient x² + x - 2 and remainder 7
Quotient x² - x + 2 and remainder -5
Using synthetic or long division with root -1 yields quotient coefficients (1,1,-2) and remainder 7. Thus the quotient is x² + x - 2 with remainder 7. .
Find the vertex of the parabola y = -2x² + 4x + 1.
(-1, -3)
(1, 3)
(1, -3)
(2, 3)
Vertex x-coordinate is -b/(2a) = -4/(2·-2) = 1. Substitute x = 1: y = -2·1 + 4·1 + 1 = 3. Thus the vertex is (1, 3). .
Simplify i?, where i is the imaginary unit.
-1
1
i
-i
Powers of i cycle every 4: i¹ = i, i² = -1, i³ = -i, i? = 1. Since 7 mod 4 = 3, i? = i³ = -i. .
Solve log?(x) + log?(x - 3) = 3.
(3 - ?41)/2
2 and -4
4 and -2
(3 + ?41)/2
Use the sum rule: log?[x(x - 3)] = 3 ? x² - 3x = 8 ? x² - 3x - 8 = 0. Solve: x = (3 ± ?(9+32))/2. Only the positive root (3 + ?41)/2 satisfies x > 3 for both logs. .
Find the sum of the first 20 terms of the arithmetic sequence with a? = 5 and d = 3.
595
670
710
690
Use S? = n/2 [2a? + (n - 1)d]: S?? = 20/2 [2·5 + 19·3] = 10[10 + 57] = 10·67 = 670. This formula applies to any arithmetic series. .
What is the range of f(x) = (x + 1)/(x - 1)?
(-?, 1) ? (1, ?)
(-?, -1) ? (-1, ?)
All real except 0
(-1, 1)
Rewrite f(x) = 1 + 2/(x - 1). As x varies, 2/(x - 1) can take any nonzero real value, so f(x) ? 1. Thus the range is all real numbers except 1. Rational function ranges.
0
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Study Outcomes

  1. Understand Key Function Concepts -

    Identify and describe domain, range, and behavior of functions commonly encountered in pre calc questions.

  2. Solve Pre Calc Equations -

    Apply algebraic techniques to solve linear, quadratic, and radical equations presented in this precalculus practice test.

  3. Apply Trigonometric Identities -

    Simplify and manipulate trigonometric expressions using fundamental identities to tackle precalculus practice problems.

  4. Analyze Graph Transformations -

    Interpret shifts, stretches, and reflections of function graphs to answer questions on the precalculus test.

  5. Evaluate Problem-Solving Strategies -

    Compare multiple approaches to complex problems and choose efficient methods during your free precalculus practice test.

  6. Interpret Performance Feedback -

    Review instant feedback to identify strengths and target areas for improvement in your precalculus practice test performance.

Cheat Sheet

  1. Function Analysis Fundamentals -

    Master domain, range, and end behavior for polynomials and rationals by following MIT OpenCourseWare techniques. For example, analyze f(x) = (x−2)/(x+3) by locating intercepts, vertical asymptotes, and sign changes using a sign chart. This strategic approach eases solving pre calc questions in your precalculus practice test.

  2. Quadratic Equations & Parabolas -

    Revise the standard form ax^2+bx+c and vertex form a(x−h)^2+k, using completing the square to find the vertex (h,k) efficiently. Apply the discriminant, b^2−4ac, from University of Utah resources to quickly determine the number of real roots on the precalculus test. Remember, (−b/2a) gives the x-coordinate of the turning point - gold for scoring high on precalc questions.

  3. Trigonometric Identities & Angle Relationships -

    Internalize key identities like sin^2θ+cos^2θ=1 and angle-sum formulas sin(α+β)=sinαcosβ+cosαsinβ from University of Colorado Boulder materials. Use the memorable phrase "Oscar Had A Heap Of Apples" (Opposite, Hypotenuse, Adjacent) for SOHCAHTOA in triangle problems. Applying these in precalculus practice problems will solidify your skills for any pre calc challenge.

  4. Exponential & Logarithmic Connections -

    Understand that e^x and ln(x) are inverses, and use the change-of-base formula log_b(a)=ln(a)/ln(b) as shown by Cornell University handouts. Practice converting growth models like P(t)=P₀e^(kt) into logarithmic form to solve for k or t swiftly. Fluency with these transformations is crucial on your precalculus practice test and daily precalc questions.

  5. Sequences, Series & Summation Techniques -

    Differentiate arithmetic sequences (a_n=a₝+(n−1)d, S_n=n/2(a₝+a_n)) from geometric ones (a_n=a₝r^(n−1), S_n=a₝(1−r^n)/(1−r)) using MIT OpenCourseWare notes. Familiarize yourself with sigma notation (∑) and practice expanding small sums to detect patterns. Mastering these formulas makes tackling precalculus practice problems and your upcoming precalculus test feel like a breeze.

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