Question

the piecewise-linear function, f, defined on ( 0 leq x leq 8 ) is shown in the graph above. the function g is given by ( g(x) = 2x + 1 ). which of the following is undefined?
( \bigcirc f(f(4)) )
( \bigcirc f(f(3)) )
( \bigcirc f(g(4)) )
( \bigcirc f(g(3)) )

Explanation:

Step1: Evaluate each option sequentially

First, identify the domain of $f(x)$: $0 \leq x \leq 8$. Any input to $f$ outside this range makes the composition undefined.

Step2: Calculate $f(f(4))$

Find $f(4)$ from the graph: at $x=4$, $f(4)=3$. Then $f(3)=2$ (from graph, $x=3$ gives $y=2$). $2$ is in $[0,8]$, so defined.

Step3: Calculate $f(f(3))$

From graph, $f(3)=2$. Then $f(2)=1.5$ (from graph, $x=2$ gives $y=1.5$). $1.5$ is in $[0,8]$, so defined.

Step4: Calculate $f(g(4))$

First compute $g(4)=2(4)+1=9$. Now check if $9$ is in $f$'s domain: $9 > 8$, so $f(9)$ is undefined. Thus $f(g(4))$ is undefined.

Step5: Calculate $f(g(3))$

Compute $g(3)=2(3)+1=7$. $7$ is in $[0,8]$, and from graph $f(7)=4.5$. So this is defined.

Answer:

$f(g(4))$