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- [Voiceover] Let's say that[br]f of x is equal to two x

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minus three, and g of x,

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g of x is equal to 1/2 x

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plus three.

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What I wanna do in this video is evaluate

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what f of g of x is,

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and then I wanna evaluate[br]what g of f of x is.

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So first, I wanna evaluate f of g of x,

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and then I'm gonna evaluate[br]the other way around.

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I'm gonna evaluate g of f of x.

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But let's evaluate f of g of x first.

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And I, like always, encourage[br]you to pause the video

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and see if you can work through it.

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This is going to be equal to,

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f of g of x is going to be equal to,

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wherever we see the x in[br]our definition for f of x,

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the input now is g of x, so[br]we'd replace it with the g of x.

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It's gonna be two times g of x.

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Two times g of x minus three.

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And this is going to[br]be equal to two times,

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well, g of x is all of that business,

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two times 1/2 x plus three,

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and then we have the minus three.

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And now we can distribute this two,

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two times 1/2 x is just[br]going to be equal to x.

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Two times three is going to be six.

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So x plus six minus three.

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This is going to equal x plus three.

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X plus three, all right, interesting.

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That's f of g of x.

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Now let's think about what[br]g of f of x is going to be.

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So g of,

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our input, instead of being,[br]instead of calling our input x,

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we're gonna call our input f of x.

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So g of f of x is going to be equal to

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1/2 times our input, which[br]in this case is f of x.

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1/2 time f of x plus three.

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You can view the x up[br]here as the placeholder

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for whatever our input happens to be.

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And now our input is going to be f of x.

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And so, this is going to[br]be equal to 1/2 times,

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what is f of x?

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It is two x minus three.

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So, two times x minus three,

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and we have a plus three.

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And now we can distribute the 1/2.

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1/2 times two x is going to be x.

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1/2 times negative three is negative 3/2s.

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And then we have a plus three.

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So let's see, three is[br]the same thing as 6/2s.

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So 6/2s minus 3/2s is going to be 3/2s.

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So this is going to be[br]equal to x plus 3/2s.

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So notice, we definitely[br]got different things

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for f of g of x and g of f of x.

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And we also didn't do a round trip.

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We didn't go back to x.

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So we know that these are[br]not inverses of each other.

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In fact, we just have to[br]do either this or that

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to know that they're not[br]inverses of each other.

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These are not inverses.

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So we write it this way.

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F of x does not equal

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the inverse of g of x.

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And g of x does not equal

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the inverse of f of x.

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In order for them to be inverses,

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if you have an x value right over here,

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and if you apply g to it,[br]if you input it into g,

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and then that takes you to g of x,

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so that takes you to g[br]of x right over here,

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so that's the function g,

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and then you apply f to it,

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you would have to get[br]back to the same place.

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So g inverse would get us back

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to the same place.

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And clearly, we did not[br]get back to the same place.

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We didn't get back to x, we[br]got back to x plus three.

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Same thing over here.

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We see that we did not get,

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we did not go get back to x,

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we got to x plus 3/2s.

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So they're definitely not[br]inverses of each other.