College Algebra, Chapter 3, 3.6, Section 3.6, Problem 24

Evaluate the expression (a) $(f \circ f) (-1)$ and (b) $(g \circ g)(2)$ using $f(x) = 3x -5$ and $g(x) = 2 - x^2$
a.) $(f \circ f) (-1)$
Solving for $f \circ f$,

$
\begin{equation}
\begin{aligned}
f \circ f &= f(f(x)) \\
\\
f \circ f &= 3(3x-5)-5 && \text{Substitute } f(x) = 3x - 5\\
\\
f \circ f &= 9x - 15 - 5 && \text{Simplify}\\
\\
f \circ f &= 9x - 20 && \text{Model}
\end{aligned}
\end{equation}
$

For $(f \circ f )(-1)$,

$
\begin{equation}
\begin{aligned}
(f \circ f )(-1) &= 9(-1) - 20 && \text{Substitute } x = -1\\
\\
(f \circ f )(-1) &= -9 -20 && \text{Simplify} \\
\\
(f \circ f )(-1) &= - 29
\end{aligned}
\end{equation}
$


b.) $(g \circ g)(2)$
Solving for $g \circ g$,

$
\begin{equation}
\begin{aligned}
g \circ g &= g(g(x))\\
\\
g \circ g &= 2-(2-x^2)^2 && \text{Substitute } g(x) = 2- x^2\\
\\
g \circ g &= 2-\left( 4 - 4x^2 + x^4 \right) && \text{Apply Distributive Property}\\
\\
g \circ g &= 2- 4 + 4x^2 - x^4 && \text{Simplify}\\
\\
g \circ g &= -2 + 4x^2 - x^4 && \text{Model}
\end{aligned}
\end{equation}
$


For $(g \circ g)(2)$,

$
\begin{equation}
\begin{aligned}
(g \circ g)(2) &= -2 + 4(2)^2 - (2)^4 && \text{Substitute } x = 2\\
\\
(g \circ g)(2) &= - 2 + 16 -16 && \text{Simplify}\\
\\
(g \circ g)(2) &= -2
\end{aligned}
\end{equation}
$