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Verify that $f$ and $g$ are inverse functions using the composition property.$$f(x)=\sqrt{9-x^{2}}, g(x)=\sqrt{9-x^{2}}$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

Missouri State University

Campbell University

Oregon State University

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Verify that $f$ and $g$ ar…

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verify that $f$ and $g$ ar…

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So they want us to show that this function route nine months expert is its own inverse. So, technically, if we do have composed with G f X, this is the same thing as G composed with f of X. So we only need to do this in one way. So let's go ahead and plug this in now. So, first, this would be the square root of nine minus F of X squared, which would be the square root of nine minus. So it would be the square root of nine minus X squared, squared to those squares. Cancel. There would be nine minus nine minus X squared, which is going to then give, um, the square root of what? The ninth council and then that just becomes X squared. But now this here is just going to become the absolute value of X. So now, technically, um, if we look at what the domain is supposed to be for this up here, well, this is going to be well, we want this to be greater than equal to zero. So that would be nine greater than equal to X squared, um, which would then give three less than or equal to X less than or equal to negative three. So we could actually have where this goes either way. So, technically, these two aren't in verses unless we restrict, uh, restrict the domain. Um, so maybe there was a typo in this where they were wanting to say, like, you know, assume X is greater than equal to zero or something like that. But without that, this actually won't be where they are in verses. So, um, kind of right off on the side. So, technically, we need X to be greater than equal to zero or X, less than or equal to zero. Uh, so I'm just going to assume it's actually the top. And if that's the case, then this is just equal to X, which would be were verified. Um, but yeah, again, we need this extra assumption to actually verify otherwise. These just aren't in verses of each other.

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