What makes two equations inverses

In this case, the domain of g x was defined as non-negative, so the absolute-value bars could be dropped above. But this isn't always the case:. Looking good so far. Now I'll plug f x into g x :. Since I started by plugging x into f x , then I was starting with any value of x. In particular, the value of x might have been negative. So g o f x does not simplify to x. The answer is: g x and f x are not inverses of each other. This is why you need to check both ways: sometimes there are fussy technical considerations, usually involving square roots, that force the composition not to work, because the domains and ranges of the two functions aren't compatible.

In this case, if f x had been restricted to non-negative x , then the functions would have been inverses. In general, though, if one composition gives you just " x ", then the other one will, too, especially if you're not dealing with restricted domains.

But you should remember to do both compositions on tests and such, in order to get full credit. Stapel, Elizabeth. Accessed [Date] [Month] The "Homework Guidelines". Study Skills Survey. We use first party cookies on our website to enhance your browsing experience, and third party cookies to provide advertising that may be of interest to you. You can accept or reject cookies on our website by clicking one of the buttons below. To understand more about how we and our advertising partners use cookies or to change your preference and browser settings, please see our Global Privacy Policy.

Page 1 of 3. So, just crunching some Algebra, here's one way to look at it: If you're got two functions, f x and g x , and then f x and g x are inverse functions.

We just need to always remember that technically we should check both. However, it would be nice to actually start with this since we know what we should get. This will work as a nice verification of the process. Now, we need to verify the results.

Here are the first few steps. Now, be careful with the solution step. With this kind of problem it is very easy to make a mistake here. Okay, this is a mess. That was a lot of work, but it all worked out in the end.

We did all of our work correctly and we do in fact have the inverse. There is one final topic that we need to address quickly before we leave this section. There is an interesting relationship between the graph of a function and the graph of its inverse.

This will always be the case with the graphs of a function and its inverse.