Piecewise functions — ACT practice questions

Piecewise functions on the ACT Mathematics test are defined by different formulas on different intervals of the domain. Students must choose the correct piece for a given input, solve for x or a parameter, and check that any solution actually belongs to that interval. Items often ask for the input that produces a stated output, the constant that makes the function continuous at a breakpoint, or the value of a composition of two such functions. Answer choices are typically numbers. A common trap is using the wrong piece or keeping a root that falls outside its stated interval.

Section: Mathematics 45 questions on the paper
45 practice questions in the app

Sample questions

Question 1

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

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Answer: B — 3

The linear branch gives $x=3$ , which does not satisfy $x\le-1$ . The quadratic branch gives $x=\pm3$ , but only $x=3$ satisfies $x>-1$ .

Question 2

The function $H$ is defined by $H(x)=cx-5$ when $x<3$ and $H(x)=4x+c$ when $x\ge3$ . For what value of $c$ is $H$ continuous at $x=3$ ?

A $\dfrac{7}{2}$
B $\dfrac{17}{2}$
C $\dfrac{23}{2}$
D17

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Answer: B — $\dfrac{17}{2}$

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

Question 3

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

$f(x)=|x|+2$ when $x\le-1$ , and $f(x)=x^2-3$ when $x>-1$ .

$g(x)=2x-1$ when $x<6$ , and $g(x)=10-x$ when $x\ge6$ .

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

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Answer: C — 6

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

Practice 45 Piecewise functions questions in the app

Piecewise functions — ACT practice questions

Sample questions

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Mathematics