Inverse Functions

All inverse functions have to be the inverse of a one-to-one functions.

A one-to-one function is a function with an output (range/ y-value) with one input (domain/ x-value).

The function above is one-to-one because the horizontal line goes through the function once.

The function above is NOT one-to-one because the horizontal line goes through the function twice.

The function above is one-to-one because the horizontal line goes through the function once.

Please note that quadratic equations are never one-to-one functions because the horizontal crosses through the function twice.

Always know that one-to-one functions always have an inverse that is a function. For example, the inverse of a quadratic is not a function because the vertical line would cross through it twice.

Please note that even functions can never be one-to-one because they one-to-one functions can never have an output with a positive and negative input. The horizontal line would have to cross the function more than once if the function is symmetrical across the y-axis.

PLEASE NOTE THIS IN THE GRAPH BELOW

Sometimes, odd functions can be one-to-one because one-to-one functions can sometimes have a positive input with positive output and a negative input with a negative output.

PLEASE NOTE THIS IN THE GRAPH BELOW

Example on Doing an Inverse of a Function:

1. ƒ(x) = 2x + 4

y=2x + 4---Change the ƒ(x) to a y.

x=2y + 4---Change the original y to an x and change the original x to a y.
-4        -4---Subtract 4 from both sides.

x - 4= 2y
  2       2---Divide by 2 from both sides.

y= x - 4
        2

ƒ⁻¹(x)= x - 4
                2---Change the y back to ƒ⁻¹(x).