Question

question suppose $h(x)=f(g(x))$. given the table of values below, determine $h(-1)$. do not include \$h(-1)=$\ in your answer.

$x$	$f(x)$	$g(x)$	$f(x)$	$g(x)$
3	-2	7	5	-1
5	-2	-2	-4	-1

Explanation:

Step1: Apply chain - rule

The chain - rule states that if $h(x)=f(g(x))$, then $h^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)$.

Step2: Evaluate at $x = - 1$

We want to find $h^{\prime}(-1)$. Substitute $x=-1$ into the chain - rule formula. When $x = - 1$, we first find $g(-1)=3$. Then $h^{\prime}(-1)=f^{\prime}(g(-1))\cdot g^{\prime}(-1)$. Since $g(-1)=3$, we need to find $f^{\prime}(3)$ and $g^{\prime}(-1)$. From the table, $f^{\prime}(3) = 5$ and $g^{\prime}(-1)=-8$.

Step3: Calculate $h^{\prime}(-1)$

$h^{\prime}(-1)=f^{\prime}(3)\cdot g^{\prime}(-1)=5\times(-8)=-40$.

Answer:

-40