Question

How do you determine whether the function `f(x)=1/x^2` has an inverse and if it does, how do you find the inverse function?

Answer 1

Simply put, the graph of `f(x) =1/x^2` has an inverse, but the inverse is not a function. In order to find the inverse of any function, interchange the `x` and `y` values and then solve for `y`.

Explanation:

In order to determine an equation of the inverse of `f(x) =1/x^2`, interchange the `x` and `y` values and then solve for `y`.

`y=1/x^2`
`x=1/y^2`
`y^2=1/x`
`y=+-sqrt(1/x)`

This is the graph of `f^-1(x) = +-sqrt(1/x)`.
graph{x=1/y^2 [-10, 10, -5, 5]}

The inverse is not a function because it does not pass the vertical line test. However, there are two methods to restrict the domain of `f(x)` so that its inverse is a function.

Method #1:

Restrict the domain of `f(x)=1/x^2` to `x>=0`. Then, the range of the inverse is `y>=0`. Since the inverse passes the vertical line test, it represents a function.

graph{y=sqrt(1/x) [-10, 10, -5, 5]}

Method #2:

Restrict the domain of `f(x)=1/x^2` to `x<=0`. Then, the range of the inverse is `y<=0`. Since the inverse passes the vertical line test, it represents a function.

graph{y=-sqrt(1/x) [-10, 10, -5, 5]}