Question

How do you verify that f(x) and g(x) are inverses: `f(x) = x+7`, `g(x) = x-7`?

Answer 1

There are two methods of checking if `f(x)` and `g(x)` are inverse functions. See explanation for details.

Explanation:

Method 1

First method is to look for inverse function of both functions.

Example.

We are looking for inverse function of `f(x)=x+7`

From the expression `y=x+7` we try to calculate `x`

`y=x+7`

`x=y-7`, so we found that `g(x)` is inverse of `f(x)`.

Now we have to look for the inverse of `g(x)`

`g(x)=x-7`

`y=x-7`

`x=y+7`

So we found that `f(x)` is the inverse function of `g(x)`

If `f` is inverse of `g` and `g` is inverse of `f` then `f` and `g` are inverse functions.

Method 2

The second way is to find the compound functions `f(g(x))` and `g(f(x))`. If they both are `h(x)=x` then `f` and `g` are inverse.

Example:

`f(g(x)=[x-7]+7` The expression in brackets is `g(x)` inserted as `x`

`f(g(x))=x-7+7=x`

`g(f(x)=[x+7]-7` The expression in brackets is `f(x)` inserted as `x`

`g(f(x))=x+7-7=x`

We found out that: `f(g(x))=g(f(x))=x`. This concludes the proof, that `f(x)` and `g(x)` are inverse functions.