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Verify Inverse Functions Using Composition

Function composition can be used to verify that two functions f (x) and g(x) are inverses.
If will be necessary to show that composing the functions, in both directions,
will yield the original input value x.

or


This confirms that the two functions will "undo" one another in either direction.

We have shown that function g(x) is the inverse of function f (x), and vice versa.

The graph of will equal x, the starting value. The graph of a function composed with its inverse function is the identity line y = x.

Determine an Invertible Function by Restricting the Domain

An invertible function is a function whose inverse is also a function. We know that a function must be a one-to-one function for the inverse to also be a function.

In the last lesson we saw an example of how a function may not have an inverse that is a function. If a inverse function is needed, the domain of the original function will need to be restricted.
Let's examine that same example, but change the question associated with the graph.

You can see that the inverse relation exists, but it is NOT a function.

Example: Determine restrictions on the function y = x2 that will allow for an invertible function to be created. While a parabola is a function, it is not a one-to-one function since it fails the horizontal line test. To be an invertible function, our restricted domain must create a one-to-one function which passes the horizontal line test. Find a portion of the graph to represent our new function. There is more than one option. For this example, cut the parabola vertically through its center (y-axis), allowing (0,0) in either portion. Working the x-axis, in either direction from the origin on the original function, will create two invertible functions. or either option creates an invertible function.

For calculator help with inverse of functions Click here.

For calculator help with inverse of functions click here.