Question

Are the following two functions inverses of one another? Use function composition to explain your answer.

Answer 1

`f(g(x)) = g(f(x)) =x`

So
`f(x)` and
`g(x)` are inverses of one other

Explanation:

If two functions
`f(x)` and
`g(x)` are inverses of each other, their compositions will equal
`x`

1. Composition
   `f(g(x))`
   Substitute the expression for
   
   `g(x) = (x-2)/(5)`
   wherever there is an
   `x` term in
   `f(x)`
   
   
   `f(x) = 5x + 2\\`
   
   So
   
   `f(g(x)) = 5 (x-2)/(5) + 2\\\\=x-2 + 2 \\\\= x`
   
2. Composition
   `g(f(x))`
   Substitute the expression for
   `f(x)` wherever there is an
   `x` term in
   `g(x)`
   
   
   `g(f(x)) = ((5x+2) - 2)/(5)\\\\= (5x+2-2)/(5)\\\\=(5x)/(5)\\\\= x`
   

Thus proved