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}
$