Piecewise function composition Practice Questions (Grade 9)

Warm-upmedium

Use the piecewise definitions of $f$ and $g$ .
$f(x)=\begin{cases} x+3 & \text{for } x \leq 3 \\ 2x-1 & \text{for } x > 3 \end{cases}$
$g(x)=\begin{cases} 4-x & \text{for } x \leq 1 \\ x+2 & \text{for } x > 1 \end{cases}$
What is $f(g(-1))\textsf{?}$

Start with the inner function: $-1 \leq 1$ , so $g(-1)=4-(-1)=5$ . Then $5 > 3$ , so $f(5)=2(5)-1=9\textsf{.}$

Mid-gamemedium

$f$ and $g$ are piecewise functions.
$f(x)=\begin{cases} 5-x & \text{for } x \leq 4 \\ 2x & \text{for } x > 4 \end{cases}$
$g(x)=\begin{cases} x+6 & \text{for } x \leq 0 \\ 3x & \text{for } x > 0 \end{cases}$
Evaluate $f(g(0))\textsf{.}$

Start with the inner function: $0 \leq 0$ , so $g(0)=0+6=6$ . Then $6 > 4$ , so $f(6)=2(6)=12\textsf{.}$

Mid-gamemedium

The functions $f$ and $g$ are defined piecewise as follows.
$f(x)=\begin{cases} |x|-1 & \text{for } x \leq 2 \\ 3x & \text{for } x > 2 \end{cases}$
$g(x)=\begin{cases} x^{2} & \text{for } x < 1 \\ 4-x & \text{for } x \geq 1 \end{cases}$
What is the value of $g(f(-4))\textsf{?}$

Start with the inner function: $-4 \leq 2$ , so $f(-4)=|-4|-1=3$ . Then $3 \geq 1$ , so $g(3)=4-3=1\textsf{.}$

Mid-gamemedium

$f$ and $g$ are defined by
$f(x)=\begin{cases} 3x & \text{for } x < 2 \\ x+1 & \text{for } x \geq 2 \end{cases}$
$g(x)=\begin{cases} x-4 & \text{for } x \leq 5 \\ 2x & \text{for } x > 5 \end{cases}$
What is $f(g(5))\textsf{?}$

Start with the inner function: $5 \leq 5$ , so $g(5)=5-4=1$ . Then $1 < 2$ , so $f(1)=3(1)=3\textsf{.}$

Mid-gamemedium

Consider the piecewise functions
$f(x)=\begin{cases} 2-x & \text{for } x \leq 0 \\ x+5 & \text{for } x > 0 \end{cases}$
$g(x)=\begin{cases} -2x & \text{for } x < 3 \\ x-1 & \text{for } x \geq 3 \end{cases}$
The value of $f(g(3))$ is

Start with the inner function: $3 \geq 3$ , so $g(3)=3-1=2$ . Then $2 > 0$ , so $f(2)=2+5=7\textsf{.}$

Mid-gamemedium

Piecewise functions $f$ and $g$ are shown.
$f(x)=\begin{cases} |x|+3 & \text{for } x \leq 4 \\ 2x-5 & \text{for } x > 4 \end{cases}$
$g(x)=\begin{cases} 6-x & \text{for } x \leq 0 \\ x+1 & \text{for } x > 0 \end{cases}$
Find the value of $f(g(0))\textsf{.}$

Start with the inner function: $0 \leq 0$ , so $g(0)=6-0=6$ . Then $6 > 4$ , so $f(6)=2(6)-5=7\textsf{.}$

Mid-gamehard

The piecewise functions $f$ and $g$ are given.
$f(x)=\begin{cases} 4x-1 & \text{for } x \leq -1 \\ 3-x & \text{for } x > -1 \end{cases}$
$g(x)=\begin{cases} x+8 & \text{for } x \leq 2 \\ 5x & \text{for } x > 2 \end{cases}$
$f(g(1))=$

Start with the inner function: $1 \leq 2$ , so $g(1)=1+8=9$ . Then $9 > -1$ , so $f(9)=3-9=-6\textsf{.}$

Buzzer beaterhard

Using these definitions of $f$ and $g$ ,
$f(x)=\begin{cases} x+2 & \text{for } x < 0 \\ 1-2x & \text{for } x \geq 0 \end{cases}$
$g(x)=\begin{cases} 3x & \text{for } x \leq 1 \\ x^{2} & \text{for } x > 1 \end{cases}$
what is $g(f(4))\textsf{?}$

Start with the inner function: $4 \geq 0$ , so $f(4)=1-2(4)=-7$ . Then $-7 \leq 1$ , so $g(-7)=3(-7)=-21\textsf{.}$

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