Question

How do you show that `f(x)=2x` and `g(x)=x/2` are inverse functions algebraically and graphically?

Answer 1

Use `f(x)` as the input for `g` and verify that this new composite function `g[f(x)]` always returns `x`.

Check that the graphs of `f` and `g` are symmetrical about `y=x`.

Explanation:

Algebraically:

Two functions are inverses when making the input of one function the output of the other creates the identity function.

Take any input `x`. Feed it to `f`, and `f` returns `f(x)`.
Now, take that `f(x)` and feed it to `g`. Then `g` returns `g[f(x)]`.
If this final output also happens to be `x`, no matter what, then `f` and `g` are inverses.

In math terms, if `f[g(x)]=g[f(x)] = x`, then `f` and `g` are inverses.

Try taking the output of `f` and feeding it to `g`:

`f(x)=2x`

`=>g(color(blue)(f(x)))=g(color(blue)(2x))=color(blue)(2x)/2=x`

So `g(f(x))=x`. In essence, whatever `f` did to `x`, the function `g` undid. Thus, `g` is the inverse of `f`.

Graphically:

Two functions are inverses when their graphs are mirror images of each other along the (identity) line `y=x`.

Here is a graph of `f(x)=2x` (the one closer to the `y`-axis) and `g(x)=x/2` (the one closer to the `x`-axis):
graph{(y-2x)(y-x/2)=0 [-10, 10, -5, 5]}

They are reflections of each other along the line `y=x`. Here is the same graph with the line `y=x` added:

graph{(y-2x)(y-x/2)(y-x)=0 [-10, 10, -5, 5]}

As you can see, `y=x` is a line of symmetry for `f` and `g`, and so `f` and `g` are inverses.