compositions of functions

Question

Use composition of functions to show that the functions f(x)=8-3x and g(x)=-1/3 x+8/3 are inverse functions. That is, carefully show that (fog)(x)=x and show that (gof)(x)=x. Show work.

Answer 1

f(g(x)) is just f(x) with g(x) substituted for x, in f:  i.e.

 

f(g(x)) = 8 - 3(-x/3 + 8/3) = 8 + x - 8 = x

 

Interpreted, if you apply g(x) to x (result 1), then apply f(x) to result 1, you get back your original x.

 

Similarly

 

g(f(x)) = -(8 - 3x)/3 + 8/3 = -8/3 + x +8/3 = x