College Algebra, Chapter 3, 3.6, Section 3.6, Problem 22

Evaluate the expression (a) $f(f(4))$ and (b) $g(g(3))$ using $f(x) = 3x -5$ and $g(x) = 2 - x^2$
a.) $f(f(4))$
Solving for $f(4)$,

$\begin{equation} \begin{aligned} f(4) &= 3(4) - 5 && \text{Substitute } x = 4\\ \\ f(4) &= 12 - 5 && \text{Simplify}\\ \\ f(4) &= 7 \end{aligned} \end{equation}$

For $f(f(4))$,

$\begin{equation} \begin{aligned} f(f(4)) &= f(7) && \text{Model}\\ \\ f(f(4)) &= 3(7) - 5 && \text{Substitute } f(4) = 7\\ \\ f(f(4)) &= 21 - 5 && \text{Simplify}\\ \\ f(f(4)) &= 16 \end{aligned} \end{equation}$


b.) $g(g(3))$
Solving for $g(3)$,

$\begin{equation} \begin{aligned} g(3) &= 2 - (3)^2 && \text{Substitute } x = 3\\ \\ g(3) &= 2 - 9 && \text{Simplify}\\ \\ g(3) &= -7 \end{aligned} \end{equation}$

For $g(g(3))$,

$\begin{equation} \begin{aligned} g(g(3)) &= g(-7) && \text{Model}\\ \\ g(g(3)) &= 2-(-7)^2 - 5 && \text{Substitute } g(3) = -7\\ \\ g(g(3)) &= 2 - 49 && \text{Simplify}\\ \\ g(g(3)) &= -47 \end{aligned} \end{equation}$