Differentiable with f(x) and and g(x) using chain rule

Question

I got this differentiable problem that I'm really stumped on.

| f(x) is the red line and g(x) is the blue line with f(x) ∗ g(x) and u(x) = f(x) g(x).

So, I'm trying to find the values of j'(-4) and u'(2). I know I need to find the slope which will be used in the chain rule of f(g(x)) = f'(g(x)) * g'(x).

I'm having trouble finding the slope of f(x) and g(x).

Greatly appreciate some help!

Answer 1

HINT

The function $f:\textbf{R}\to\textbf{R}$ is given by \begin{align*} f(x) := \frac{2x}{3} + \frac{2}{3} \end{align*}

The function $g:\textbf{R}\to[1/2,2]$ is given by \begin{align*} g(x) := \begin{cases} 1, & x\in(-\infty,-2],\\\\ \dfrac{x}{2} + 2, & x\in(-2,0],\\\\ -\dfrac{x}{2} + 2& x\in(0,3],\\\\ \dfrac{1}{2} & x\in(3,+\infty). \end{cases} \end{align*}

Now you can obtain the expression of $f(g(x))$.

Can you take it from here?