Question

the table above gives values of f, f, g, and g for selected values of x. if h(x)=f(g(x)), what is the value of h(1)?

Explanation:

Step1: Recall chain - rule

The chain - rule states that if $h(x)=f(g(x))$, then $h^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)$. We want to find $h^{\prime}(1)$. First, we need to find $g(1)$ and $g^{\prime}(1)$.

Step2: Find $g(1)$ from the table

From the table, when $x = 1$, $g(1)=3$.

Step3: Find $g^{\prime}(1)$ from the table

From the table, when $x = 1$, $g^{\prime}(1)=-2$.

Step4: Find $f^{\prime}(g(1))$

Since $g(1)=3$, we need to find $f^{\prime}(3)$. From the table, when $x = 3$, $f^{\prime}(3)=7$.

Step5: Calculate $h^{\prime}(1)$

Using the chain - rule $h^{\prime}(1)=f^{\prime}(g(1))\cdot g^{\prime}(1)$. Substitute $g(1) = 3$ and $g^{\prime}(1)=-2$ and $f^{\prime}(3)=7$ into the formula. So $h^{\prime}(1)=f^{\prime}(3)\cdot g^{\prime}(1)=7\times(-2)=-14$.

Answer:

$-14$