Study on the fly ACT MathematicsPiecewise functions

Piecewise functions — ACT practice questions

Mathematics
45 questions on the paper
45 practice questions in the app
Free

Practice 45 Piecewise functions questions in the app

What this topic is

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.

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 $v(x)=2x+7$ when $x\le -1$ 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 $x\le-1$ . The quadratic branch gives $x=\pm3$ , but only $x=3$ satisfies $x>-1$ .

Question 2Mid

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 $x=3$ ?

\dfrac{7}{2}

✗ Not this one

\dfrac{17}{2}

✓ Correct

\dfrac{23}{2}

✗ Not this one17✗ Not this one

Show answer

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 3Mid

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 $f(g(2))$ ?

-1

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

Show answer

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 45 Piecewise functions questions in the app

Piecewise functions — ACT practice questions

What this topic is

Sample questions

More ACT topics

Mathematics