You have the following function:
[tex]f(x)=2x-8[/tex]In order to determine f^-1, replace x by y and f(x) by x and solve for y, as follow:
[tex]\begin{gathered} x=2y-8 \\ x+8=2y \\ \frac{x+8}{2}=y \end{gathered}[/tex]As you can notice, the obtained expression for y is the inverse function.
Hence:
f^-1(x) = (x + 8)/2
In order to verify the given identities, proceed as follow:
[tex]\begin{gathered} (f\circ f^{-1})(x)=f(f^{-1}(x)) \\ =f(\frac{x+8}{2}) \\ =2(\frac{x+8}{2})-8 \\ =x+8-8 \\ =x \end{gathered}[/tex]Now, for the other identity:
[tex]\begin{gathered} (f^{-1}\circ f)(x)=f^{-1}(f(x)) \\ =f^{-1}(2x-8) \\ =\frac{2x-8+8}{2} \\ =\frac{2x}{2} \\ =x \end{gathered}[/tex]