Algebra: Inverse Functions, Finding Inverses

I'm so sorry this post is a week late. Life got busy. :-)

Now that we understand the definition of an inverse function and how to test them, we can learn how to find them. In this post, I'll share my favorite method of finding the inverse. How do you like to do it?

Step 1: Rewrite with the variables y and x. Use y in place of f(x), g(x), etc.

This is straight-forward. If you are given a function in terms of different variables, such as f(b) and b, change them to y and x. Use y for the dependent variable (usually f(x) or g(x), etc.) and x for the independent variable (usually already x, but may be another letter).

Step 2: Swap the independent and dependent variables.

Swap x and y. Shown below.

Step 3: Solve for y.

Using this new equation, solve for y. Shown below.

Step 4: Replace y with the inverse notation of the original function.

In this case, we replace y with f^(-1)(x) because the original function was f(x). If the original function were, say, g(b), then we would replace x with b, and y with g^(-1)(b). Shown below.

Step 5: Test the inverse.

Using the method we discussed in the previous post, test the inverse you found. Those steps are shown below.

Our answer passes the test of the theorem we learned about, so it is correct.

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Let's try another example. This one is a little more complicated, and will show you why it is important to test your answer at the end!

Steps 1-3: Rewrite in terms of y and x; swap the variables; solve for y.

All three of the first steps are shown below. Notice the last step of solving for y. When you take the square root of a variable, you must include both the positive and negative outside of the square root. Therefore, we actually have two answers for the inverse, this time.

Step 4: Rewrite with inverse notation of the original function.

Step 5: Test the inverse(s).

We must test both of our answers. First, we will test the positive version, shown below.

This answer passes the test. Now, we must test the negative version, shown below.

This answer passes the first portion of the test, but fails the second portion. Remember, both parts of the theorem must hold true in order for the two functions to be considered inverses. Therefore, g^(-1)(x) = - sqrt(x-1) is NOT an inverse of g(x) = x^2 + 1! We simply throw out this answer and only give the positive version, shown below.

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That's how I like to find the inverse of a function. :-) If you have another method you prefer, I'd love to hear all about it in a comment!

That concludes this algebra series on inverse functions. If you have a request for my next series, leave it as a comment on this, or any, post, or send it to me via e-mail (find my e-mail on the Contact page). I'd love to hear from you!

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!

Interesting Solution

A reader, Richard Odland, contacted me with a different approach to puzzle I shared some time ago, "A Corny Question" on this post. (My solution can be found on the follow-up, Solutions post.)

I found his approach interesting, and received his permission to share it with you all!

Here is a link to a Google Document with his solution. :) Let us know what you think of it!

And thank you, Richard!