Question

question suppose h(x)=f(g(x)). given the table of values below, determine h(2). do not include \h(2)=\ in your answer.

x	f(x)	g(x)	f(x)	g(x)
2	6	3	0	-4
3	4	6	5	-4
5	4	4	-2	3

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 $x = 2$

We want to find $h^{\prime}(2)$. Substitute $x = 2$ into the chain - rule formula. When $x = 2$, $g(2)=3$ and $g^{\prime}(2)=-4$. Then $h^{\prime}(2)=f^{\prime}(g(2))\cdot g^{\prime}(2)$. Since $g(2)=3$, we need to find $f^{\prime}(3)$. From the table, $f^{\prime}(3)=5$.

Step3: Calculate $h^{\prime}(2)$

$h^{\prime}(2)=f^{\prime}(3)\cdot g^{\prime}(2)$. Substitute $f^{\prime}(3)=5$ and $g^{\prime}(2)=-4$ into the equation. So $h^{\prime}(2)=5\times(-4)=-20$.

Answer:

-20