College Algebra, Chapter 3, 3.6, Section 3.6, Problem 34

Determine the functions $f \circ g, \quad g \circ f, \quad f \circ f$ and $g \circ g$ and their domains if $f(x) = 6x - 5$ and $\displaystyle g(x) = \frac{x}{2}$
For $f \circ g$

$\begin{equation} \begin{aligned} f \circ g &= f(g(x)) && \text{Definition of } f \circ g\\ \\ f \circ g &= 6 \left( \frac{x}{2} \right) - 5 && \text{Definition of } g\\ \\ f \circ g &= 3x - 5 && \text{Definition of } f \end{aligned} \end{equation}$

The domain of the function is $(-\infty, \infty)$

For $g \circ f$

$\begin{equation} \begin{aligned} g \circ f &= g(f(x)) && \text{Definition of } g \circ f\\ \\ g \circ f &= \frac{6x-5}{2} && \text{Definition of } g\\ \\ g \circ f &= 3x - \frac{5}{2} && \text{Definition of } f \end{aligned} \end{equation}$

The domain of the function is $(-\infty,\infty)$

For $f \circ f$

$\begin{equation} \begin{aligned} f \circ f &= f(f(x)) && \text{Definition of } f \circ f\\ \\ f \circ f &= 6(6x-5)- 5 && \text{Definition of } f\\ \\ f \circ f &= 36x - 30 - 5 && \text{Simplify}\\ \\ f \circ f &= 36x - 35 && \text{Definition of } f \end{aligned} \end{equation}$

The domain of the function is $(-\infty,\infty)$

For $g \circ g$

$\begin{equation} \begin{aligned} g \circ g &= g(g(x)) && \text{Definition of } g \circ g\\ \\ g \circ g &= \frac{\frac{x}{2}}{2} && \text{Definition of } g\\ \\ g \circ g &= \frac{x}{4} && \text{Definition of } g \end{aligned} \end{equation}$

The domain of the function is $(-\infty,\infty)$