1.5 Inverse FUNctions

What is an Inverse Function?

To every action there is an equal and opposite reaction. This logic, or Newton’s third law of motion, can also be applied to Mathematical Functions. Any function you can think of has a corresponding and opposite Function or Relation, called Inverse Functions/Relations. These functions/relations undo the original function. It’s a misconception that every function has an inverse one. While every function has an opposite mathematical procedure that undoes it, not every one of these inverses is a function, meaning some of them have multiple outputs for just one input, making it an Inverse Relation. For those who aren't fluent in mathematical lexicon, the Inverse function flips the domain and the range, so whatever was originally the output of a certain input, is then made an input, whose output was its input in the original function. To further demonstrate this idea, here is what the equations of a Function and its Inverse would look like 

Function:

Inverse Function:

How to identify Inverse Functions:

For a Function to have an Inverse Function, it must be a One-to-one Function. One-to-one Functions are functions with exactly one corresponding output for each input. The name ‘One-to-One’ is derived from the equation that defines f(a)=f(b), which is then simplified to a=b.

How to write an Inverse Function:

To find the Inverse Function, you must use the Additive Inverse(-x) and Multiplicative Inverse(1/x)

 the 22 becomes negative and is added to the 'x', and then it's all divided by 12

Inverse Functions & Their Graphs:

An Inverse Function's graph is similar to the original functions graph, in that it is reflected over the line y=x, and it MUST pass the Vertical Line Test and the Horizontal Line Test.

Although the original function in the graph above has been reflected over y=x, it is not an example of an Inverse Function. The graph does NOT pass both the Vertical and Horizontal Line Test.

The above graph is a perfect example of a Function and it's Inverse. It's been reflected over y=x and passes both the Vertical and Horizontal Line Test.