Composite Functions

Question

The functions f and g are as follows :

f(x) = 1 (x+1)² g(x) = 1/x

A) Find g⁻¹(x)

B) gf(x)

C) Solve for gf(x)=16

x = __ or x = __

Answer 1

g(x) = 1/x

g^-1(x) = 1/x

g(x) = g^-1(x)

y = 1/x switch x and y and solve for the new y to find the inverse function

x = 1/y

y = 1/x

y' = 1/x = f^-1(x)

f(x) = 1(x+1)^2 = (x+1)^2

g(f(x)) = g((x+1)^2) = 1/(x+1)^2

f(g(x)) = (1/x +1)^2