Inverse function

Inverse functions are a way to "undo" a function. In the original function, plugging in x gives back y, but in the inverse function, plugging in y (as the input) gives back x (as the output). If a function were to contain the point (3,5), its inverse would contain the point (5,3).

If the original function is f(x), then its inverse f ^-1(x) is not the same as .

To find the inverse of a function, you need to do the opposite of what the original function does to x.

Example

Not all functions have inverses. A function must be a one-to-one function, meaning that each y-value has a unique x-value paired to it. Basically, the same y-value cannot be used twice. The horizontal line test can determine if a function is one-to-one. Imagine finding the inverse of a function that is not one-to-one. Plugging in a y-value from the original function could return more than one x-value. The inverse function would not be a function anymore.

If a function is not one-to-one, you will need to apply domain restrictions so that the part of the function you are using is one-to-one.

To find the inverse of a function, you can use the following steps:

1. In the original equation, replace f(x) with y:

to

2. Replace every x in the original equation with a y and every y in the original equation with an x

Note: It is much easier to find the inverse of functions that have only one x term. For functions that have more than one x term, you will need to solve for y by moving all y terms to one side of the equation and factoring out a y.

Example

3. Solve for y:

(1)

(2)

(3)

(4)

(5)

4. Change y to f ^-1(x):

5. Apply domain restrictions as necessary.

Because the inverse of a function will return x when you plug in y, the range of the original function will be the domain of its inverse. Similarly, the domain of the original function will be the range of its inverse.

When the original function is not one-to-one, you will need to restrict its domain so that it is one-to-one, then look at the range from that part of the function.

In this case, you know that the range of the original function, , is [-3, ∞). Therefore, the domain of the inverse function, , will be [-3, ∞) as well.

Let's look at the graph of the original function and its inverse:

If you notice, the inverse function (red) is a reflection of the original function (blue) across the line y = x. This is true for all functions and their inverses.

You can also check that you have the correct inverse function beecause all functions f(x) and their inverses f ^-1(x) will follow both of the following rules:

(f ∘ f ^-1)(x) = x

(f ^-1 ∘ f)(x) = x

Note: The "∘" symbol indicates composite functions. Essentially, function composition involves applying one function to the results of another. Refer to the composite functions page for further detail or a refresher on composite functions.

The reason that the above rules are true is because a function and its inverse are reflections of each other over the line y = x. Going back to our example, we can check if we got the right inverse function using these rules.

Recall the following:

and

Comparing (f ∘ f ^-1)(x) and (f ^-1 ∘ f)(x), we see that:

Since the result in both cases is x, this confirms that we found the correct inverse.

See also composite functions.