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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 $f\left(f^{-1}(x)\right)=x$.$$f(x)=\sqrt{2 x-8}$$

(a) $x \geq 4$(c) $y \geq 0$(f) $x \geq 0$$(g) y \geq 4$(h) $f^{-1}(x)=\left(x^{2}+8\right) / 2$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

Campbell University

Oregon State University

Harvey Mudd College

Baylor University

Lectures

06:35

In Exercises $29-38,$ for …

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06:32

08:30

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01:16

In Problems $29-38,$ the f…

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In Exercises $31-36,$ (a) …

02:29

04:08

03:58

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In Exercises $23-30,$ (a) …

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So in order for us to find the domain of this here or what will need to do is think. Well, we know we can't take the square root of negative numbers. So we're just going to set them inside of here greater than zero and then solve. So this is going to be two X might say, strictly larger or greater than or equal to zero. And then we add a divide to that gives us X is greater than or equal to four. So that is going to be our domain. Or if you want, you can write it as for including it to infinity. So either one of these ways is about way to write the domain. Uh, now to graph this so let's go ahead and put that down. Uh, So actually, before I do this, let's go ahead in first graph or at least put some points down for the regular square root function. And then we can use our transformations to throw out what the new graph is going to be. Um, so we know we normally have 00 on there. Um, actually, maybe I should put this is like a chart. So X square root effects. I mean, you could also just plug this into, like, a calculator or something, but I assume we're just supposed to be doing this all by hand, so we plug in zero, we get out. Zero. If we plug in one, we get out. One the next perfect square is four. So that would be too. And I think this is going to be good enough for what we're doing. So now if we go ahead and look at what transformations we have over here Well, this as we shift eight to the right and then we compress by two or do you think of it as divide by two? So, uh, the Y values are still going to stay the same. In this case, it's just the X values that are going to be moving. So x Route two x he works by a seat is going to be so it's still be 01 and two. Those didn't change. But then again, remember, the X values are going to be changing, so zero. So we take that so we add eight to it. Then we divide by two. So that would be four. Uh, and then one, we add eight and then we divide by two. So that would be 9 to 4.5. And then here we add eight. So that would be 12. And then we divide by two. And that would be six. Yes, we have that. So I have to go bit out. It looks like so 12345678 And then we just need to go up to two on the Y axis of 12 All right, so now, um, So we start at 40 that we have 4.51 and then we have six two. So this is what we have or our function here. So this is why is it to f of X now? We can actually go ahead and graph f inverse. So I'll just go ahead and do that while I'm at it. Because, remember, all we're going to do is if this is X y, then these points over here are supposed to be y x. So we can just take these points that we have and then switch them around. So 40 so x f inverse of x. So this becomes 04 This becomes one 4.5, and then this becomes 26 Actually, let me go ahead and scooped this down a bit and let me erase that. Okay, then dispute. This will damage it. Do that. So to put that back there and then skewed all of this down, Okay? Uh huh. So, yeah, affects there. And now we need to go up to at least six. So 123456 So we can go ahead and plot those points of 04 And we have 14.5 and then to six. And so this is going to be Why is ego to f inverse of X? And one way to kind of confirm this is the case. Remember if we did the line, why is equal to X here? So this wasn't asked, but we have wisely. Good X knows how. These are reflections across this. Um, yeah, so that's just kind of a quick way of going about it. So now the range for F of X, which was our green one over here. Um, well, knows how it starts from zero and then just keeps on going up. So this is going to be from zero to infinity. Yeah. Now to show this is 1 to 1, there's a couple of things we could do. Um, we could just look at the green and say, Well, it passes horizontal line test but I wouldn't say that's too interesting. What I would do instead is say, Well, let's look at the derivative of this. Um So if we look at the derivative so f prime of X, Well, remember, this is really like it was to a one half power, so we would use power rules. It would be one half b two x minus eight, raised to the negative one half. And then we take the derivative on the inside, do the chain rule, which would be times too. So those cancel out and then this is going to be 1/2 X minus eight and actually let me scoot down. I'm probably also music that done as well so that square root of this. So now notice that this is going to be strictly larger than zero, which implies this is always increasing. So since it's always increasing, we know that this is one of the cases of where our function is going to be 1 to 1. All right, now, to find the range of the domain of the inverse, all we need to do is come over here to our domain and range of our original function, and then switch them. So the domain of F inverse is going to be our range. So this is zero to infinity, and then the range of our inverse is just going to be our domain, which is, or to infinity. And if you also look at what the red graph is, you can see how that is the case as well. Um, next, we need to find what f in verses. So I should let me take this and bring it down here. So remember, the first thing we're going to do is go ahead and switch this out for why? And then we're gonna interchange X and y. So this is going to be X is equal to square root of two Y minus X, and then we're going to solve for y. So we're going to square. Each side that gives us X squared is equal to two y minus eight. Add the aid over so X squared. Plus eight. Is it good to To why divide each side by two and that's going to give. Why is he two x squared? Plus 8/2, which is going to be f adverse effects so you could go ahead and divide the two. But I'm just going to kind of leave it like this. Okay, so we have that now. Um But there's one thing that we should make note of, because knows here this is defined for all values. So off on the side, it would probably be a good idea for us to say that X is greater than or equal to. Um Well, our domain was supposed to be zero. So this is the actual true inverse of it? Um, yeah, it's important that you say greater than or equal to zero. Otherwise, um, knows if we come up here and look at our graph, we only have, like, one half of a quadrant we don't have Where it's like also coming up like this. Yeah, So just something to kind of keep in mind. Um, next, they want us to show that the composition both ways gives us just x, so I'll go ahead and start with f inverse of f A bex. Yeah, So we're going to take ffx and plug it into f inverse. So this is going to be the square root of two X minus eight. Well, actually, let me just plug in ffx first So f of X squared plus 8/2. All right now we can go ahead and plug that in. So it would be the square root of two X minus eight squared plus eight all over to so first notice square square root, cancel out. And then the negative eight and the eight cancel and I will leave us with two x over two, which is equal to just X because those twos also can't solve with each other. So this way checks out. Now let's do it the other way around. So it b f f f inverse of x. So we're going to take f inverse and plug it into F and doing it this way. We'll need to be careful, and I'll kind of explain why. So we have the square root of two x minus not to x to f in verse of X minus eight and then we can go ahead and plug in f inverse. So this would be the square root of two X Square plus 8/2 minus eight. So first notice these two's cancel and then we just have X squared. Plus eight mice ate the AIDS cancel, and we're going to be left with the square root of X squared. Now, this is not X, but this is actually just the absolute value of X. And to make sure that this is just equal to X, So this is going to be equal to X because notice that our domain on this inside function f inverse if we come up here and look again, it's from zero to infinity. So this is going to just be equal to X, since the domain of F inverse is equal to 02 infinity. Um, so this is something a lot of times people kind of forget about, but it is something that is important to make note of because otherwise just doing this composition normally results in the absolute value of X as opposed to just X

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