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ACT Math Help: Piecewise Functions

Review real example questions for Piecewise Functions in ACT Math.

Question 1 / 10

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Which interval contains $x = -1$ for the piecewise function $f(x) = \begin{cases} 3x + 7 & \text{if } x \leq -1 \\ x^2 - 2 & \text{if } x > -1 \end{cases}$ ?

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Question 1

Which interval contains $x = -1$ for the piecewise function $f(x) = \begin{cases} 3x + 7 & \text{if } x \leq -1 \\ x^2 - 2 & \text{if } x > -1 \end{cases}$ ?

$x < -1$
$x \leq -1$ (correct answer)
$x > -1$
$x = 0$

Explanation: To determine which interval contains $x = -1$ , we check each condition: Is $-1 \leq -1$ ? Yes. Is $-1 > -1$ ? No. Since $-1$ satisfies the condition $x \leq -1$ , it belongs to the first interval. The boundary point $x = -1$ is included in the first piece due to the $\leq$ symbol.

Question 2

What is $f(0)$ for the piecewise function $f(x) = \begin{cases} 2x + 4 & \text{if } x < 1 \\ x^2 - 6 & \text{if } x \geq 1 \end{cases}$ ?

4 (correct answer)
2
6
0

Explanation: For x = 0, we check the intervals: Is 0 < 1? Yes. So we use the first piece: $f(x) = 2x + 4$ . Substituting x = 0: $f(0) = 2(0) + 4 = 0 + 4 = 4$ . Choice B would result from using the second piece incorrectly.

Question 3

A savings plan applies a rule $f(x)$ to the number of weeks $x$ you have saved. For the piecewise function $f(x)=\begin{cases}6-x & \text{if } x<4\\ 2x+1 & \text{if } 4\le x<9\\ x^2-10 & \text{if } x\ge 9\end{cases}$ what is $f(9)$ ?

19
81
8
71 (correct answer)

Explanation: For x = 9, determine which piece to use: Is 9 < 4? No. Is 4 ≤ 9 < 9? No, since 9 is not less than 9. Is 9 ≥ 9? Yes. Use the third piece: f(x) = x² - 10. Thus f(9) = 9² - 10 = 81 - 10 = 71.

Question 4

Which interval contains x = 3 for the function $$f(x) = \begin{cases} 3x + 1 & \text{if } x < 1 \ 2x - 2 & \text{if } 1 \leq x < 4 \ x^2 & \text{if } x \geq 4 \end{cases}

$x < 1$
$1 \leq x < 4$ (correct answer)
$x \geq 4$
$x > 4$

Explanation: For $x = 3$ , check each interval: $3 < 1$ ? No. $1 \leq 3 < 4$ ? Yes, since $1 \leq 3$ and $3 < 4$ . $3 \geq 4$ ? No. Therefore, $x = 3$ falls in the interval $1 \leq x < 4$ .

Question 5

What is f(2) for the piecewise function: $$f(x) = \begin{cases} -x + 3 & \text{if } x < 1 \ 4x & \text{if } 1 \leq x < 3 \ x^2 - 1 & \text{if } x \geq 3 \end{cases}

8 (correct answer)
7
9
6

Explanation: For x = 2, check intervals: 2 < 1? No. 1 ≤ 2 < 3? Yes. So use the second piece $f(x) = 4x$ . Substitute x = 2: $f(2) = 4(2) = 8$ . The value x = 2 falls clearly within the middle interval.

Question 6

A company assigns a performance rating $f(x)$ based on an employee's score $x$ . The rating function is

7-x & \text{if } x<0 \\ 3x+1 & \text{if } 0\le x<4 \\ 15 & \text{if } x\ge 4 \end{cases}

Based on the piecewise function, what is the value when $x=0$ ?

7
1 (correct answer)
0
15

Explanation: For x = 0, check intervals: Is 0 < 0? No. Is 0 ≤ 0 < 4? Yes, since 0 = 0 satisfies this condition. Use the second piece: f(x) = 3x + 1. Substituting: f(0) = 3(0) + 1 = 0 + 1 = 1. The boundary x = 0 falls in the middle piece due to the ≤ sign.

Question 7

A game assigns points $f(x)$ based on a player's level $x$ using the piecewise function below. For

3x+2 & \text{if } x<2\\ 10 & \text{if } 2\le x<5\\ -x+20 & \text{if } x\ge 5 \end{cases}

what is $f(2)$ ?

8
10 (correct answer)
12
18

Explanation: For x = 2, check which interval contains 2: Is 2 < 2? No. Is 2 ≤ 2 < 5? Yes, since 2 ≤ 2 is true and 2 < 5 is true. Therefore, use the second piece f(x) = 10. Since this piece is a constant function, f(2) = 10. The boundary x = 2 belongs to the middle interval due to the ≤ sign.

Question 8

A machine's output $f(x)$ depends on the setting $x$ using the piecewise function below. For

-x+6 & \text{if } x<1\\ 2x & \text{if } 1\le x<6\\ x^2-10 & \text{if } x\ge 6 \end{cases}

what is $f(0)$ ?

0
6 (correct answer)
12
-6

Explanation: For x = 0, check intervals: Is 0 < 1? Yes. Therefore, use the first piece f(x) = -x + 6. Substituting x = 0: f(0) = -0 + 6 = 6. Since 0 is less than 1, we don't need to check the other intervals.

Question 9

A grading policy assigns a score adjustment $f(x)$ based on the raw score $x$ . For the piecewise function

-x & \text{if } x<-1\\ 2x+5 & \text{if } -1\le x<3\\ 11 & \text{if } x\ge 3 \end{cases}

what is $f(-1)$ ?

3 (correct answer)
1
7
11

Explanation: For x = -1, check intervals: Is -1 < -1? No. Is -1 ≤ -1 < 3? Yes, since -1 ≤ -1 is true and -1 < 3 is true. So use the second piece f(x) = 2x + 5. Substituting x = -1: f(-1) = 2(-1) + 5 = -2 + 5 = 3. The boundary x = -1 belongs to the middle interval due to the ≤ sign.

Question 10

Which interval contains $x = -2$ for the piecewise function $f(x) = \begin{cases} x^2 & \text{if } x \leq -2 \\ 3x + 5 & \text{if } x > -2 \end{cases}$ ?

$x \geq -1$
$x > -2$
$x < -3$
$x \leq -2$ (correct answer)

Explanation: To determine which interval contains $x = -2$ , we check each condition: Is $-2 \leq -2$ ? Yes. Is $-2 > -2$ ? No. Since $-2$ satisfies the condition $x \leq -2$ , it belongs to the first interval. The boundary point $x = -2$ is included in the first piece due to the $\leq$ symbol.