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Current time:0:00Total duration:3:33

- [Instructor] In this video,
we're gonna think about function inverses a little bit more, or whether functions are
inverses of each other, and specifically we're gonna 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 three, and let's say that g of x
is equal to the square root, the principal root of x minus three. 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. And 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
one, what is f of one? Well, it's gonna be
one squared plus three. That's one plus three, that is four. So if g is an inverse of f,
then if I input four here, I should get one. Now, that wouldn't prove
that their 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 four here,
four minus three is one. The principal root of that is one, so that's looking pretty good. Let's try one more value here. Let's try two. Two squared plus three is seven. Now let's try out seven here. Seven minus three is four. The principal root of that is two. So, so far it is looking pretty good. But then what happens if
we try a negative value? Pause the video and think about that. Let's do that. Let me put a negative two right over here. Now, if I have negative two
squared, that's positive four, plus three is seven, so I have seven here. But we already know that
when we input seven into g, we don't get negative two, we get two. In fact, there's no
way to get negative two 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 could 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,
that using specific points, you 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, and 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 four x, and let's say that j of x
is equal to x over four. We know that these are
inverse of each other. We'll prove it in other
ways in future videos, but you can't try every single input here and look at 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.