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Inverse Functions

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- Inverse Functions and Notation
- Reverse Operations to Write Inverse Functions
- The Graph of a Function and its Inverse

**Inverse Functions and Notation**

In a function, the input variable undergoes one or more operations to yield a specific output value. We can think of this ideas as x being the input to f(x), where the output value is y. We can say that f(x) = y.

With inverse functions, the operations performed on the input variable is undone. This means that when y is put into the inverse function, the result will be the x value that produced y in the original function.

We can write this using inverse notation as:

The above expression is telling us that if the value of a function, f(x), goes into an inverse function, f^{-1}(x), the result will be the argument of the original function, x.

**Reverse Operations to Write Inverse Functions**

We can write an inverse function by applying inverse operations in reverse order, according to operations that make up a function. Consider the following example:

Let's work backwards, and apply inverse operations. This will lead to our inverse function:

**The Graph of a Function and its Inverse**

Below is the graph of and

Notice that there is symmetry between the graph of a function and its inverse. The line y = x acts as a line of reflection between a function and its inverse. This is characteristic of all functions and their inverses.

The line y = x is a line of reflection between a function and its inverse. This means that if you have the graph of a function, you can also graph the line y = x, and use this line to plot points on the inverse function.