Using specific values to test for inverses | Precalculus | Khan Academy

In this video, we're going to think about function inverses a little bit more, or whether functions are inverses of each other. Specifically, we're going to think about can we tell that by essentially looking at a few inputs for the functions and a few outputs.

So, for example, let's say we have f of x is equal to x squared plus 3, and let's say that g of x is equal to the square root, the principal root of x minus 3. Pause this video and think about whether f and g are inverses of each other.

All right, now one approach is to try out some values. So, for example, let me make a little table here for f. So this is x, and then this would be f of x. Then let me do the same thing for g, so we have x, and then we have g of x.

Now, first let's try a simple value. If we try out the value 1, what is f of 1? That's going to be 1 squared plus 3. That's 1 plus 3; that is 4. So if g is an inverse of f, then if I input 4 here, I should get 1. Now that would prove that they're inverses. But if it is an inverse, we should at least be able to get that.

So, let's see if that's true. If we take 4 here, 4 minus 3 is 1. The principal root of that is 1. So that's looking pretty good. Let's try one more value here. Let's try 2. Two squared plus three is seven. Now let's try out seven here. Seven minus three is four; the principal root of that is 2.

So, so far it's looking pretty good. But then what happens if we try a negative value? Pause the video and think about that. So, let's do that. Let me put a negative 2 right over here.

Now, if I have negative 2 squared, that's positive 4 plus 3 is 7. So I have 7 here. But we already know that when we input 7 into g, we don't get negative 2; we get 2. In fact, there's no way to get negative 2 out of this function right over here.

So, we have just found a case, and frankly, any negative number that you try to use would be a case where you can show that these are not inverses of each other. Not inverses.

So, you actually can use specific points to determine that two functions like this, especially functions that are defined over really an infinite number of values, these are continuous functions, can show examples where they are not inverses. But you actually can't use specific points to prove that they are inverses because there's an infinite number of values that you could input into these functions. There's no way that you're going to be able to try out every value.

For example, if I were to tell you that h of x, really simple functions, h of x is equal to 4x, and let's say that j of x is equal to x over 4. We know that these are inverses of each other. We'll prove it in other ways in future videos, but you can't try every single input here and every single output, and every single input here and every single output.

So we need some other technique, other than just looking at specific values, to prove that two functions are inverses of each other. Although you can use specific values to prove that they are not inverses of each other.