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Radical and rational functions — ACT practice questions

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Practice 38 Radical and rational functions questions in the app

What this topic is

Domain restrictions are the defining feature of Radical and rational functions.

A square root function is defined only where the expression under the root is zero or positive, and a rational function is undefined wherever its denominator equals zero. When a factor cancels from both top and bottom, the graph has a hole at that x-value instead of a vertical asymptote.

For example, the fraction with x minus one times x plus four on top and x minus one times x plus three on the bottom has a hole at x equals one and a vertical asymptote at x equals negative three. Choices describe domains, ranges or graph behavior.

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 $h (x) = \frac{(x - 2)^{1 / 2}}{x - 6}$ ?

x \geq 2

✗ Not this one

x \geq 2

and

x \neq 6

✓ Correct

x > 2

and

x \neq 6

✗ Not this oneAll real

x

except 6✗ Not this one

Show answer

Answer: B — $x \geq 2$ and $x \neq 6$

The radicand requires $x - 2 \geq 0$ , so $x \geq 2$ , and the denominator is zero at $x = 6$ , which must be excluded. The domain is $x \geq 2$ with $x \neq 6$ .

Question 2Mid

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

The domain is

[- \frac{5}{4}, \infty)

, 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

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

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Practice 38 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