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Verify that $f$ and $g$ are inverse functions using the composition property.$$f(x)=x^{2}+1, x \geq 0, g(x)=\sqrt{x-1}$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

Missouri State University

Oregon State University

Lectures

02:04

Verify that $f$ and $g$ ar…

01:06

Use function composition t…

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02:30

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01:34

verify that $f$ and $g$ ar…

01:23

For the following exercise…

02:50

for the following exercise…

So if we want to confirm that these are in Versant by composition, the first thing we're going to do is look at composed with G of X. And we want to see if this is just X, and then we're going to do in the opposite way as well. So, uh, this is going to be g of X squared plus one, which would give us the square root of X minus one squared plus one. Uh, so that would just be expense one plus one. The ones cancel and we're just left with X. So that checks out now for the other way G of f of X. This is going to be the square root of F A bex minus one. So if we plug that in now, the X squared plus one minus one, the ones cancel. We get square root of X squared, which is just going to be the absolute value of X. So, uh, if we look to see what our domain is well, on the inside here it's FX. So for the most part, that is just what our domain is going to be. So if we come over here and look well, that's supposed to be greater than equal to zero. So then that means this here is just going to be X. And then you can say, since domain of F is greater than or equal to zero. But again, we've shown that this composition, no matter how we do it, just leaves us with X. So they are in versus

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