So the first one is g(x)=(x/2)+1
Find g^-1 [g(2)]
Second is h(x)=(x/3)+1
Find h[h^-1 (x)]
first note that these functions are one to one and that from:
$\displaystyle g(x) = \frac{x}{2} \ + \ 1$
$\displaystyle g(2) = \frac{2}{2} \ + \ 1 \ = 2$
to get the inverse temporarily change this equation to
$\displaystyle y = \frac{x}{2} \ + 1$
now exchange x for y and solve for y
$\displaystyle x = \frac{y}{2} + 1$
$\displaystyle y = 2x - 2 \ or \ g^{-1}(x) = 2x - 2$
$\displaystyle then\ g^{-1}[g(2)] = 2(2) - 2 \ = \ 2$
now as for $\displaystyle h(x) = \frac{x}{3} + 1$
$\displaystyle h^{-1}(x) = 3x -3$
so $\displaystyle h[h^{-1}(x)] = \frac{3x-3}{3} - 3 = x-4$
You don't need to do all the calculation bigwave did (the first is correct and nicely done- for the second he found h^-1(x), not h[h^-1(x)]) and it really doesn't matter how g and h are defined (as long as they have inverses).
By the definition of inverse, g^-1[g(2)]= 2 and h[h^-1(x)]= x!
If f is any function having an inverse and a, b is any numbers in the domains of f and f^-1 respectively, f^-1(f(a))= a and f(f^-1(a))= a. That's the definition of "inverse function".