Question

Consider the following functions
f(x)= (7x+8)/(x+3) and g(x)= (3x-8)/(7-x)
(a) Find g(g(x))
(b) Find g(f(x))
(c) Determine whether the functions f and g are inverses of each other.

Answer 1

a)
g(g(x))
= g((3x-8)/(7-x)
= [3(3x-8)/(7-x) - 8]/( 7 - (3x-8)/(7-x) )
= [ (9x-24-56+8x)/(7-x)] / (49-7x-3x+8)/(7-x)]
= (17x - 80)/(57 - 10x)

b) do this in the same way I just showed you

c) let's take the inverse of f(x)
let y = (7x+8)/(x+3)
for the inverse, interchange the x and y
x = (7y+8)/(y+3)
xy + 3x = 7y + 8
xy - 7y = 8-3x
y(x-7) = 8-3x
y = (8-3x)/(x-7) = (3x - 8)/(7 - x) = g(x)

yes , they are inverses of each other.