1.5 Inverse Functions

What is an inverse?

Graphically, an inverse of a function is the reflection of that function over the line y = x. This translates into algebra as a function that undoes another function.

This picture shows the function f(x) reflected in the line y=x to make the inverse function f^-1(x).

Notation:

The inverse of a function f is denoted by f^-1(read f-inverse).

Does every function have an inverse?

No! Not every function has an inverse! In order for a function to have an inverse it must be one-to-one, meaning there is only one y coordinate for every x coordinate.

To find out graphically if a function has an inverse, you must do the horizontal line test.  If a horizontal line crosses the graph at more than one point for any spot that it is placed, then the function does NOT have an inverse.

This function does not have an inverse that exists as a function because it is not one-to-one.











Algebraically, a function does not have an inverse when

A function does have an inverse if and only if,

meaning that no two y-coordinates have the same x-coodinate, unless those coordinates are equal.


How do you find a function's inverse?

To find the inverse of a function, you must switch the x and y coordinates and then solve for the new y.  This will undo everything done in the equation, which by definition will leave you with the inverse function.


Additional Properties of Inverse Functions

Two functions f and g are inverses of each other if

and

Links

A helpful, but long video http://www.youtube.com/watch?v=nSmFzOpxhbY

A shorter helpful video http://www.youtube.com/watch?v=wSiamij_i_k