Verifying inverse functions by composition: not inverse | High School Math | Khan Academy
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Verifying inverse functions by composition: not inverse | High School Math | Khan Academy

October 11, 2019


– [Voiceover] Let’s say that
f of x is equal to two x minus three, and g of x, g of x is equal to 1/2 x plus three. What I wanna do in this video is evaluate what f of g of x is, and then I wanna evaluate
what g of f of x is. So first, I wanna evaluate f of g of x, and then I’m gonna evaluate
the other way around. I’m gonna evaluate g of f of x. But let’s evaluate f of g of x first. And I, like always, encourage
you to pause the video and see if you can work through it. This is going to be equal to, f of g of x is going to be equal to, wherever we see the x in
our definition for f of x, the input now is g of x, so
we’d replace it with the g of x. It’s gonna be two times g of x. Two times g of x minus three. And this is going to
be equal to two times, well, g of x is all of that business, two times 1/2 x plus three, and then we have the minus three. And now we can distribute this two, two times 1/2 x is just
going to be equal to x. Two times three is going to be six. So x plus six minus three. This is going to equal x plus three. X plus three, all right, interesting. That’s f of g of x. Now let’s think about what
g of f of x is going to be. So g of, our input, instead of being,
instead of calling our input x, we’re gonna call our input f of x. So g of f of x is going to be equal to 1/2 times our input, which
in this case is f of x. 1/2 time f of x plus three. You can view the x up
here as the placeholder for whatever our input happens to be. And now our input is going to be f of x. And so, this is going to
be equal to 1/2 times, what is f of x? It is two x minus three. So, two times x minus three, and we have a plus three. And now we can distribute the 1/2. 1/2 times two x is going to be x. 1/2 times negative three is negative 3/2s. And then we have a plus three. So let’s see, three is
the same thing as 6/2s. So 6/2s minus 3/2s is going to be 3/2s. So this is going to be
equal to x plus 3/2s. So notice, we definitely
got different things for f of g of x and g of f of x. And we also didn’t do a round trip. We didn’t go back to x. So we know that these are
not inverses of each other. In fact, we just have to
do either this or that to know that they’re not
inverses of each other. These are not inverses. So we write it this way. F of x does not equal the inverse of g of x. And g of x does not equal the inverse of f of x. In order for them to be inverses, if you have an x value right over here, and if you apply g to it,
if you input it into g, and then that takes you to g of x, so that takes you to g
of x right over here, so that’s the function g, and then you apply f to it, you would have to get
back to the same place. So g inverse would get us back to the same place. And clearly, we did not
get back to the same place. We didn’t get back to x, we
got back to x plus three. Same thing over here. We see that we did not get, we did not go get back to x, we got to x plus 3/2s. So they’re definitely not
inverses of each other.

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