How to Find the Inverse of a Function

Step 1: Make Sure the Function is one-to-one

A function will have an inverse only if it is one-to-one. So before you attempt to find the inverse of a function, check this first. Recall that a function f is one-to-one if every element y in the range corresponds to exactly one element x in the domain. A quick way to see that a function is one-to-one is to use the horizontal line test.

Once you know that the function has an inverse, you can follow a few simple steps to find it. We’ll use the following function as our model:
f(x)=3x/(x+1)

Step 2: Change the “ f(x)” to “y”
For this first step, you’re simply changing the variable’s symbol.
y=3x/(x+1)

Step 3: Switch “x” and “y”
This means that you change every x you see to y, and change every y to x. You’ll usually only have one y initially. But you may, as in our case, start with more than one x.
x=3y/(y+1)

Step 4: Solve for y
This step is just solving an equation. It may take some time, depending on the complexity of the function.
x=3y/(y+1) multiply by (y+1)

x(y+1)=3y distribute the x

xy+x=3y move y-terms to one side

x=3y-xy factor out the y

x=y(3-x) divide by (3-x)

y=x/(3-x)

Step 5: Change the “y” to “f^-1(x)”
For this last step, just change the variable’s symbol back. Remember to use f^-1(x), instead of f(x), to represent the inverse of f(x).
f^-1(x)=x/(3-x)

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