College Algebra Practice Test

Review functions, polynomials, rational expressions, exponential and logarithmic models, sequences, and conics.

College Algebra Practice Test

20 college-level algebra questions with worked explanations.

Instant feedback · Worked explanations

College Algebra Structure Maproom

College algebra is less about isolated procedures and more about how functions, theorems, and graphs fit together.

A college-level review moves beyond routine manipulation. Domain restrictions must survive simplification, composition is evaluated from the inside outward, inverse functions reverse the input-output relationship, polynomial theorems replace longer computations, and graph features such as asymptotes and transformations must agree with algebraic form. The most reliable approach is to identify the structure first and then use the shortest theorem or representation that exposes it.

DomainsCompositionInverse functionsPolynomial zerosRemainder TheoremDiscriminantsRational asymptotesExponentialsLogarithmsArithmetic seriesGeometric seriesTransformationsCirclesPolynomial construction

1. Domain restrictions belong to the original function, not just the simplified form

Rational functions exclude denominator zeros; radical functions require a nonnegative radicand when using real values.

Rational domain

f (x) = \frac{x^{2} + 1}{x - 4}

x \neq 4

The denominator cannot equal zero, so the excluded input is recorded before any other work.

Radical domain

g (x) = \sqrt{7 - x}

7 - x \geq 0

x \leq 7

The radicand condition determines the allowed real inputs.

2. Composition and inversion are different operations with different logic

Composition substitutes one function into another; inversion swaps input and output and then solves for the new output.

Composition from the inside outward

f (x) = 2 x + 3

g (x) = x^{2} - 1

f (g (x)) = 2 (x^{2} - 1) + 3

f (g (x)) = 2 x^{2} + 1

The inner function is evaluated first. Reversing the order generally changes the result.

Inverse function

y = \frac{2 x - 5}{3}

x = \frac{2 y - 5}{3}

3 x = 2 y - 5

y = \frac{3 x + 5}{2}

f^{- 1} (x) = \frac{3 x + 5}{2}

An inverse function reverses the original relation; it is not the reciprocal of the original formula.

3. Polynomial theorems can replace full division or direct root solving

The Remainder Theorem evaluates once; known zeros construct factors immediately.

Remainder Theorem

x^{3} - 4 x^{2} + x + 6

x - 2

f (2) = 8 - 16 + 2 + 6

f (2) = 0

A zero remainder confirms the linear factor without polynomial long division.

Construct a polynomial from zeros

x = 2

x = - 3

f (x) = (x - 2) (x + 3)

f (x) = x^{2} + x - 6

Each zero becomes a factor whose root reproduces that zero.

4. The discriminant classifies quadratic roots without solving the quadratic

The sign of the discriminant tells whether the quadratic has two real roots, one repeated real root, or no real roots.

Quadratic

2 x^{2} + 3 x + 5 = 0

Compute the discriminant

D = 3^{2} - 4 \cdot 2 \cdot 5

D = - 31

Interpret the sign

D < 0

A negative discriminant means the quadratic has no real roots.

5. Rational-function asymptotes come from different parts of the formula

A vertical asymptote comes from a forbidden denominator value; a horizontal asymptote describes long-run behavior.

Read asymptotes from structure

f (x) = \frac{2}{x - 3} + 1

x \neq 3

x = 3

y = 1

A numerator zero would describe an intercept, not a vertical asymptote. The denominator controls the vertical break.

6. Exponential and logarithmic equations use inverse relationships, not linear rules

Match exponential bases when possible, and keep logarithm domains active after combining expressions.

Exponential equation

5^{x - 1} = 125

5^{x - 1} = 5^{3}

x = 4

Equal positive bases let the exponents be compared directly.

Logarithmic equation

\log_{2} (x - 1) + \log_{2} (x - 3) = 3

x > 3

\log_{2} ((x - 1) (x - 3)) = 3

(x - 1) (x - 3) = 8

x^{2} - 4 x - 5 = 0

(x - 5) (x + 1) = 0

x = 5

The second algebraic candidate is rejected because it violates the original logarithm domain.

7. Arithmetic and geometric series use different accumulation formulas

Arithmetic series add terms with constant differences; geometric series add terms generated by repeated multiplication.

Arithmetic series

S_{n} = \frac{n}{2} (2 a_{1} + (n - 1) d)

a_{1} = 4, d = 3, n = 8

S_{8} = 4 (8 + 21)

S_{8} = 116

Geometric series

S_{n} = a_{1} \cdot \frac{1 - r^{n}}{1 - r}

a_{1} = 3, r = 2, n = 5

S_{5} = 3 \cdot \frac{1 - 2^{5}}{1 - 2}

S_{5} = 93

Choosing the arithmetic formula for multiplicative growth is a structural error, not just an arithmetic one.

8. Function transformations are easier to interpret graphically than by expansion alone

Horizontal and vertical shifts move the graph without changing its basic shape.

Base function and translated function

f (x) = x^{2}

g (x) = {(x - 2)}^{2} + 3

(2, 3)

The translated vertex shows the horizontal and vertical shifts directly.

9. Standard conic form turns an equation into geometric information

For a circle, the translated squared terms identify the center, while the right side gives the squared radius.

Read a circle from standard form

{(x - 2)}^{2} + {(y + 1)}^{2} = 25

(2, - 1)

r = 5

{(x - h)}^{2} + {(y - k)}^{2} = r^{2}

The sign inside each translated square is opposite the corresponding coordinate of the center.

10. Logarithm laws apply to products and quotients, not to sums

A common college-algebra error is treating a sum inside a logarithm as though it were a product.

Tempting but invalid

\log (a b)

\log (a) + \log (b)

A sum inside the logarithm does not split into two logarithms.

Keep the distinction explicit

\log (a b) \neq \log (a) + \log (b)

The addition rule students often remember does not exist.

Product rule

\log (a b) = \log (a \cdot b)

Products, not sums, separate into sums of logarithms.

11. College-algebra errors usually come from confusing similar-looking structures

The correct procedure depends on the exact operation and representation in front of you.

Inverse function confused with reciprocal

An inverse reverses input and output; a reciprocal simply places an expression in a denominator.

Composition order reversed

The inner function must be evaluated first.

Excluded denominator value forgotten

Domain restrictions belong to the original function and survive cancellation.

Numerator zero called a vertical asymptote

Vertical asymptotes come from denominator behavior after common-factor analysis.

Exponential equation treated as linear

The variable appears in an exponent, so inverse or base-matching methods are needed.

Logarithm rule applied to a sum

Product and quotient laws do not distribute across ordinary addition.

Final college-algebra audit

Before accepting an answer, confirm the domain, operation order, theorem conditions, graph behavior, and whether the final candidate survives the original restrictions.

What domain restrictions exist before simplifying?Record denominator zeros and radical conditions immediately.

What is the operation order?Composition works from the inside outward; inversion swaps input and output.

Is there a theorem that shortens the work?Use the Remainder Theorem, discriminant, or series formulas when the structure matches.

Does the graph agree with the formula?Check asymptotes, transformations, center, radius, and root behavior.

Are logarithm and exponential rules being used on the correct operations?Products and powers behave differently from sums.

Does every final answer satisfy the original domain?Verification belongs to the original equation, not only the transformed one.

This block supports the College Algebra Practice Test. Its examples are illustrative rather than copies of the test questions. The focus is domains of rational and radical functions, composition and inverse functions, polynomial zeros and the Remainder Theorem, discriminants, rational-function asymptotes, exponential and logarithmic equations, arithmetic and geometric series, function transformations, circles, standard conic form, and polynomial construction from zeros.