Piecewise functions — ACT practice questions

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Practice 46 Piecewise functions questions in the app

What this topic is

Piecewise functions use different rules on different parts of the domain, so the first step is always deciding which rule applies to the given input.

For instance, if f(x) equals x plus one when x is negative and equals two times x when x is zero or positive, then f of negative four uses the first rule and f of three uses the second. Solving an equation means checking each rule separately and discarding solutions outside that rule's interval.

ACT Mathematics also asks for the value of a constant that makes the pieces meet, called continuity, and for composite values. Answer choices often reflect using the wrong piece.

Sample questions

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

Question 1Mid

The function $v$ is defined by $v (x) = 2 x + 7$ when $x \leq - 1$ and $v (x) = x^{2} + 4$ when $x > - 1$ . For what value of $x$ is $v (x) = 13$ ?

- 3

✗ Not this one3✓ Correct6✗ Not this one9✗ Not this one

Show answer

Answer: B — 3

The linear branch gives $x = 3$ , which does not satisfy $x \leq - 1$ . The quadratic branch gives $x = \pm 3$ , but only $x = 3$ satisfies $x > - 1$ .

Question 2Mid

The function $H$ is defined by $H (x) = c x - 5$ when $x < 3$ and $H (x) = 4 x + c$ when $x \geq 3$ . For what value of $c$ is $H$ continuous at $x = 3$ ?

\frac{7}{2}

✗ Not this one

\frac{17}{2}

✓ Correct

\frac{23}{2}

✗ Not this one17✗ Not this one

Show answer

Answer: B — $\frac{17}{2}$

Continuity requires the two branch expressions to agree at $x = 3$ . Setting $3 c - 5 = 12 + c$ gives $2 c = 17$ , so $c = \frac{17}{2}$ .

Question 3Mid

The functions $f$ and $g$ are defined as follows.

$f (x) = ∣ x ∣ + 2$ when $x \leq - 1$ , and $f (x) = x^{2} - 3$ when $x > - 1$ .

$g (x) = 2 x - 1$ when $x < 6$ , and $g (x) = 10 - x$ when $x \geq 6$ .

What is the value of $f (g (2))$ ?

- 1

✗ Not this one5✗ Not this one6✓ Correct8✗ Not this one

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Answer: C — 6

Because $2 < 6$ , $g (2) = 2 (2) - 1 = 3$ . Because $3 > - 1$ , $f (3) = 3^{2} - 3 = 6$ .

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

Practice 46 Piecewise functions questions in the app

Piecewise functions — ACT practice questions

What this topic is

Sample questions

More ACT topics

Mathematics