## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

$f[g(x)]=g[f(x)]=x$. This means that $f(x)$ and $g(x)$ are inverses of each other. Both $f(x)$ and $g(x)$ are continuous on the set of real numbers, so there are no restrictions to their domains.
We wish to plug $f(x)$ into $g(x)$ to obtain: $$\displaystyle g[f(x)]=\frac{1}{2} (2x+6)-3 \\=x+3-3 \\=x$$ We wish to plug $g(x)$ into $f(x)$ to obtain: $$f[(g(x)]=2 (\dfrac{x}{2}-3)+6 \\=x-6+6 \\=x$$ We see that $f[g(x)]=g[f(x)]=x$. This means that $f(x)$ and $g(x)$ are inverses of each other. Both $f(x)$ and $g(x)$ are continuous on the set of real numbers, so there are no restrictions to their domains.