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Radical and rational functions — ACT practice questions

Mathematics
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Practice 37 Radical and rational functions questions in the app

What this topic is

Radical and rational functions on the ACT Mathematics test cover square roots, other roots, and quotients of polynomials, including where those expressions are defined and how they graph.

Students must find the real-number domain of a function, describe both domain and range, and interpret a rational graph at a canceled factor, such as a hole rather than a vertical asymptote. Questions are multiple choice and often ask which statement describes the domain, the range, or the graph at a given x-value.

Common traps include treating a removable discontinuity as an asymptote, forgetting that even roots require a nonnegative radicand, and including values that make a denominator zero.

Sample questions

Pick an answer to see whether it is right — nothing is saved, and nothing needs an account.

Question 1Mid

Which of the following describes the real-number domain of the function $h(x)=\dfrac{(x-2)^{1/2}}{x-6}$ ?

x\ge 2

✗ Not this one

x\ge 2

and

x\ne 6

✓ Correct

x>2

and

x\ne 6

✗ Not this oneAll real

x

except 6✗ Not this one

Show answer

Answer: B — $x\ge 2$ and $x\ne 6$

The radicand requires $x-2\ge0$ , so $x\ge2$ , and the denominator is zero at $x=6$ , which must be excluded. The domain is $x\ge2$ with $x\ne6$ .

Question 2Mid

Given that $f(x)=\sqrt[3]{4x+5}$ , which statement correctly describes the domain and range of $f$ ?

The domain is

\left[-\dfrac{5}{4},\infty\right)

, and the range is all real numbers.✗ Not this oneThe domain is all real numbers, and the range is all real numbers.✓ CorrectThe domain is all real numbers, and the range is

[0,\infty)

.✗ Not this oneThe domain excludes

-\dfrac{5}{4}

, and the range excludes 0.✗ Not this one

Show answer

Answer: B — The domain is all real numbers, and the range is all real numbers.

A cube root is defined for every real radicand and can produce every real output. The linear expression inside does not restrict either set.

Question 3Mid

Which of the following statements best describes the graph of $y=\dfrac{(x-2)(x+5)}{(x-2)(x+1)}$ at $x=2$ ?

There is a hole at

x=2

.✓ CorrectThere is a vertical asymptote at

x=2

.✗ Not this oneThe graph crosses the

x

-axis at

x=2

.✗ Not this oneThe graph has a horizontal asymptote

y=2

.✗ Not this one

Show answer

Answer: A — There is a hole at $x=2$ .

Canceling $(x-2)$ leaves $\dfrac{x+5}{x+1}$ for $x\neq 2$ , so $x=2$ is a removable discontinuity (hole), not an asymptote.

Every question is picked for where you are right now, wrong answers come back until they stick, and it all works offline.

Practice 37 Radical and rational functions questions in the app

Radical and rational functions — ACT practice questions

What this topic is

Sample questions

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Mathematics