Question

given the graphs of (y = f(x)) and (y = g(x)) shown below and (h(x)=f(x)g(x)), determine the value of (h(-7)).

Explanation:

Step1: Recall the product - rule

The product - rule states that if $h(x)=f(x)g(x)$, then $h^{\prime}(x)=f^{\prime}(x)g(x)+f(x)g^{\prime}(x)$. We need to find $f(-7)$, $f^{\prime}(-7)$, $g(-7)$ and $g^{\prime}(-7)$ from the graphs.

Step2: Find $f(-7)$ and $g(-7)$ from the graphs

Looking at the graphs of $y = f(x)$ and $y = g(x)$, when $x=-7$, for the function $y = f(x)$, the point on the graph gives $f(-7)= 9$. For the function $y = g(x)$, when $x = - 7$, $g(-7)=-2$.

Step3: Find $f^{\prime}(-7)$ and $g^{\prime}(-7)$

The slope of the line segment of $y = f(x)$ near $x=-7$: The line segment of $y = f(x)$ has two - point $( - 7,9)$ and $( - 5,5)$. The slope $m_f=\frac{5 - 9}{-5+7}=\frac{-4}{2}=-2$, so $f^{\prime}(-7)=-2$. The slope of the line segment of $y = g(x)$ near $x=-7$: The line segment of $y = g(x)$ has two - point $(-7,-2)$ and $(-5,-4)$. The slope $m_g=\frac{-4 + 2}{-5 + 7}=\frac{-2}{2}=-1$, so $g^{\prime}(-7)=-1$.

Step4: Apply the product - rule

Substitute the values into the product - rule formula $h^{\prime}(x)=f^{\prime}(x)g(x)+f(x)g^{\prime}(x)$. When $x=-7$, $h^{\prime}(-7)=f^{\prime}(-7)g(-7)+f(-7)g^{\prime}(-7)$. Substitute $f^{\prime}(-7)=-2$, $g(-7)=-2$, $f(-7)=9$ and $g^{\prime}(-7)=-1$ into the formula: $h^{\prime}(-7)=(-2)\times(-2)+9\times(-1)=4 - 9=-5$.

Answer:

$-5$