Algebra: Inverse Functions, Testing Inverses

Remember that theorem I gave you in the Introductory post for this series?

We can use this theorem to test whether two functions are inverses of each other.

To test whether these two functions are inverses, we need to look at the two composite functions in the theorem above.

Step 1: Evaluate the first composite function in the theorem above, (f o g)(x).

To form a composite function, locate the variable (x) in the first function (circled in red in the image above). Insert the entire second function into this variable (circled in green, above, with arrow showing the insertion point).

Now, evaluate this new function, shown below.

If the first composite function equals x, continue to Step 2.

Otherwise, the two functions are NOT inverses.

In this example, the first composite function, (f o g)(x), does equal x. Therefore, we continue to Step 2.

Step 2: Evaluate the second composite function in the theorem above, (g o f)(x).

Form this composite function as before. This time, the entire first function (circled in red above, with arrow showing insertion point) is inserted into the variable placeholder of the second function (circled in green).

As before, evaluate this new function, shown below.

If the second composite function equals x, the functions ARE inverses.

If EITHER of the two composite functions do not equal x, the functions ARE NOT inverses!

In this example, the two functions are inverses of each other.

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Here's another example.

Begin with Step 1 as before.

The first composite function cannot be reduced any further than (3x^2 - 1), as shown in the image above. Therefore, it does not equal x. And therefore, the functions f(x) and g(x) are NOT inverses.

There is no need to proceed to Step 2. According to the theorem, both composite functions must equal x for the two functions to be inverses of each other.

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The composite functions need not be evaluated in the order given. You could evaluate the second composite function first. Sometimes, one of the two is easier to reduce than the other. If you get stuck trying to reduce one of them, try the other.

In the next, and final, post of this series, we'll discover how to find the inverse of a given function. :-) I hope you'll stick around for it!