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?

1 Answer
Brandon Y.
Nov 29, 2017

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]}

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