Problem
For each pair of functions $f$ and $g$ below, find $f(g(x))$ and $g(f(x))$. Then, determine whether $f$ and $g$ are inverses of each other. Simplify your answers as much as possible. (Assume that your expressions are defined for all $x$ in the domain of the composition. You do not have to indicate the domain.) (a) $f(x)=\frac{x+5}{2}$ (b) $f(x)=2 x$ \[ g(x)=2 x-5 \] \[ g(x)=2 x \] \[ f(g(x))=\square \] \[ \begin{array}{l} f(g(x))=\square \\ g(f(x))=\square \end{array} \] \[ g(f(x))=\square \] \[ g(f(x))=\square \] $f$ and $g$ are inverses of each other $f$ and $g$ are inverses of each other $f$ and $g$ are not inverses of each other $f$ and $g$ are not inverses of each other
Solution
Step 1 ：\(f(g(x))=f(2x-5)=\frac{(2x-5)+5}{2}=\frac{2x}{2}=x\)
Step 2 ：\(g(f(x))=g(\frac{x+5}{2})=2(\frac{x+5}{2})-5=x+5-5=x\)
Step 3 ：\(\boxed{\text{Since } f(g(x))=x \text{ and } g(f(x))=x, \text{ f and g are inverses of each other.}}\)
Step 4 ：\(f(g(x))=f(2x)=2(2x)=4x\)
Step 5 ：\(g(f(x))=g(2x)=2(2x)=4x\)
Step 6 ：\(\boxed{\text{Since } f(g(x))=4x \text{ and } g(f(x))=4x, \text{ f and g are not inverses of each other.}}\)
