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In Exercises $29-38,$ for the function determined by the given equation (a) determine its domain, (b) sketch its graph, (c) determine its range, (d) show the function is one-to-one, (e) sketch the graph of its inverse function, (f) determine the domain of the inverse function, (g) determine the range of the inverse function, (h) find the equation of the inverse function and (i) verify that $f^{-1}(f(x))=x$ and $f\left(f^{-1}(x)\right)=x$.$$f(x)=x^{2}, x \geq 0$$

(a) $x \geq 0$(c) $y \geq 0$(f) $x \geq 0$$(g) y \geq 0$(h) $f^{-1}(x)=\sqrt{x}$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

Oregon State University

Harvey Mudd College

Baylor University

Lectures

08:30

In Exercises $29-38,$ for …

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In Problems $29-38,$ the f…

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In Exercises $31-36,$ (a) …

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In Exercises $23-30,$ (a) …

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So if we want to find the domain of this here, they actually tell us what it is. They're just saying X is going to be greater than you go to zero So we can write that at zero to infinity. The reason why we could just go ahead and throw that down is because remember, X, where it is a quadratic or more general terms of polynomial and Paul. No meals are defined for all values. Next, they want us to graph this. So let's go ahead and do that. So 12341234 So I'll just put down the first three points. Um, which would be 0011 and then 24 So this here is going to be Why is equal to F a bex? Now if we want to, um, get the inverse, we can do that without even finding it, because, remember, we know that if we have, like, X y here on the inverse, this just gets taken to y X. So if we graph these points again or just write them down 00 11 and 24 then these are going to get taken to Well, that's still 00 That's still 11 And then this is going to be four to, So we'd have 00 11 and then, uh, 42 And so we know this will look something kind of like that, and that's going to be Y is equal to f inverse of X. So we have both of those on the same graph Now, Now, for our range, um, we could just go ahead and look at the green because this was supposed to be just for a fax, and so this would just also be zero to infinity. Now we want to show that this is going to be 1 to 1. So we could just look at the Green Line and say, Well, it passes the horizontal line test or we can look at the derivative. So if we come up here and take the derivative so the f prime of X, um and that is going to just be powerful. So that's two X, but we have the little restriction of X being greater than or equal to zero. So now if we just come down here and multiply this by two, that is going to say two X is greater than equal to zero, which is going to imply that F Prime of X is going to be greater than or equal to zero, which implies it is always and increasing. So that's how we can use the fact, um, that this is always increasing early, such as, say, on X greater than or equal to zero, which is where we're interested in. Okay, uh, now, to get our range in our domain of our inverse, we don't even need to look at the graph up here. Since we already have our domain and range of our original function, all we need to do is flip them around. So the domain of F in verse is going to be the range of F, which would be zero to infinity. And then the range of F inverse is supposed to be the domain of F, which is going to just be zero to infinity. So each case is going to be the same thing, just kind of by chance. Now they want us to find the inverse. Let me go ahead and scoop this down. So remember the first thing we're going to do is replaced this with Hawaii, and then we're going to interchange. So this is going to be X is equal to why squared, And then we're just going to take the square root on each side. So we actually need to be a little careful when we do this. The square root square root Because this is going to actually give us that X is equal to the absolute value of why the square root here and now, in order for us to drop this absolute value, we need to go and look and see Well, what is supposed to be the range of our inverse, Because why you're supposed to be that? And well, here it goes from zero to infinity, so we can just go ahead and drop the absolute value. Uh, so this is going to be the square root of X, actually, me, uh, just right. Like this square root of X is equal to y since range of F inverse is equal to zero to infinity because otherwise we get, like, plus or minus here. Um, yeah. So we have that, and then that is going to imply. So I'll just kind of write this over here. That f inverse of X is going to be equal to the square root of X. So that is our inverse. Yeah. Okay, Now we want to show that no matter how we compose these, we still get just X. So if we have f inverse of f of X, so we'll go ahead and plug X square in. So this would be f so f inverse square root of f of X, which is going to be the square root of expert. And then again, this is the absolute value of X. But if we look at what the domain of alphabets Wiz, which we said was zero to infinity, this then is just going to be X. And then we can say, since the domain of F is 02 infinity. Because remember, when we do this composition, we start on the inside four. Our domain. Um, yeah. So, again, this is important to make this kind of, uh, statement because otherwise it could be the negative as well if we were on a different domain. Um, yeah. So that checks out now for the other way around. Um, f of f inverse of X. Yeah, so that would be f inverse of X. But we square it so that we plug that into the square root of X squared. And then here. That's just going to be equal to X because we don't have an absolute value when we do it in the other way. Yeah, so that one checks out so you can see how we get both of them being just x.

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