Question

How do you find the inverse of `f(x)=1/x^2`?

Answer 1

`y=+-1/sqrtx`

Explanation:

Rewrite as `y=1/x^2`.

Switch the `x` and `y`.

`x=1/y^2`

Solve for `y`.

`xy^2=1`

`y^2=1/x`

`y=+-1/sqrtx`

Notice that the inverse is not actually a function, but two coexisting functions: `1/sqrtx` and `-1/sqrtx`.

You can tell if a function's inverse will be a function if the function is one-to-one, or if it passes the "horizontal line test".

This is the graph of `f(x)`:
graph{1/x^2 [-10.25, 9.75, -2.04, 7.96]}

Notice how all `y`-values are represented by two `x`-values. Since the domain and range of the inverse function will flip, you can tell the inverse won't be a function.