Explanation:
a regular function finds the functional result y to a given x value.
the inverse function finds the original x to a given y value.
g^-1(1) is therefore the x-value, so that g(x) = 1.
so, we look through the value pairs. where do we find y = 1 ? ah, in the pair (9, 1).
x = 9 lead to the functional result y = 1.
therefore, g^-1(1) = 9
h(x) = y = -3x - 14
in other words, h(x) expresses y in terms of x.
h^-1(x) expresses x in terms of y.
so, we want to transform the functional equation, so that it says "x = ...." :
y = -3x - 14
y + 14 = -3x
x = (y + 14)/-3
and to turn it into regular function notation, we rename x to y and y to x :
y = h^-1(x) = (x + 14)/-3 = (-1/3)(x + 14)
(h○h^-1)(x) = x
always.
in that process we find first what original input value for h lead to the value of x. and then we use that as input value for h. and, of course, we have to get x as result.
to check here in our case
(h○h^-1)(-5) = -5
h^-1(-5) = (-5 + 14)/-3 = 9/-3 = -3
h(-3) = -3×-3 - 14 = 9 - 14 = -5
there you have it.