If [tex]f^{-1}(x)[/tex] is the inverse of [tex]f(x)[/tex], then by definition of inverse functions,
[tex]\left(f\circ f^{-1}\right)(x) = f\left(f^{-1}(x)\right) = x[/tex]
Given [tex]f(x) = 2x-8[/tex], the inverse, if its exists, satisfies the equation above. Evaluate [tex]f[/tex] at the inverse, we have
[tex]f\left(f^{-1}(x)\right) = 2 f^{-1}(x) - 8 = x[/tex]
Solve for the inverse:
[tex]2 f^{-1}(x) - 8 = x[/tex]
[tex]2 f^{-1}(x) = x + 8[/tex]
[tex]\boxed{f^{-1}(x) = \dfrac12 x + 4}[/tex]
Given [tex]g(x) = \frac13 x + 5[/tex], we do the same thing as before to find its inverse.
[tex]g\left(g^{-1}(x)\right) = \dfrac13 g^{-1}(x) + 5 = x \implies \boxed{g^{-1}(x) = 3x - 15}[/tex]
