Question

How would you show that `f (x) = 7x +3` and `f^-1(x) = (x +3 )/ 7` are inverses of each other?

Answer 1

Verify that `f^(-1)(x) = (x-3)/7` not `(x+3)/7`

Explanation:

These functions are not inverses of one another.

The correct formula for `f^(-1)(x)` can be written:

`f^(-1)(x) = (x-3)/7`

In fact the variable name used does not matter, so I will write:

`f^(-1)(y) = (y-3)/7`

Then substituting `y=f(x)` we find:

`f^(-1)(f(x)) = (f(x)-3)/7 = ((7x+3)-3)/7 = (7x)/7 = x`

Substituting `x=f^(-1)(y)` we find:

`f(f^(-1)(y)) = 7f^(-1)(y)+3 = 7((y-3)/7)+3 = (y-3)+3 = y`