Question

1. the graph of the function h is shown below.

graph of h
if f is the function given by $f(x) = h(h(x))$, what is the value of $f(1)$?

Explanation:

Step1: Recall chain rule for derivatives

For $f(x)=h(h(x))$, the chain rule gives $f'(x)=h'(h(x))\cdot h'(x)$

Step2: Find $h(1)$ from the graph

From the graph, at $x=1$, $h(1)=3$

Step3: Find $h'(1)$ (slope at $x=1$)

The left segment has slope: $\frac{2-4}{2-0}=\frac{-2}{2}=-1$, so $h'(1)=-1$

Step4: Find $h'(3)$ (slope at $x=3$)

The right segment has slope: $\frac{6-2}{4-2}=\frac{4}{2}=2$, so $h'(3)=2$

Step5: Compute $f'(1)$

Substitute into chain rule: $f'(1)=h'(h(1))\cdot h'(1)=h'(3)\cdot h'(1)$
$=2\times(-1)$

Answer:

$-2$