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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 $f\left(f^{-1}(x)\right)=x$.$$f(x)=2 x-3$$

(a) $(-\infty, \infty)$(c) $(-\infty, \infty)$(f) $(-\infty, \infty)$$(g)(-\infty, \infty)$(h) $f^{-1}(x)=(x+3) / 2$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

Missouri State University

Oregon State University

University of Michigan - Ann Arbor

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06:28

In Exercises $29-38,$ for …

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In Problems $29-38,$ the f…

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In Exercises $23-30,$ (a) …

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So we first want to determine Well, what is the domain of two X minus three? And, well, the domain of this, Um, since this is polynomial or we just think about it the line as well. We know there is no bad numbers to plug into here. So the domain of this is just going to be from negative infinity to infinity. Um, Then we'll come over here to be so I'm gonna kind of do these a little bit out of order just because it's easier for me to kind of group them this way. So let's come over here and put our graph down first, and we want to graph f of X two X minus three So we know that, uh, we should have our y intercept at negative three, actually, just put down a bunch of lines everywhere. So we have a negative three, and then the slope here says we go up to for everyone unit we go over, go up to for everyone unit, we go over and then we can just go ahead and connect these here. So this is going to be why is equal to X. Okay. Uh, now, if we want to find the range of this. We can go ahead and just really look at what our graph is and see that Well, this goes from negative infinity to infinity as well. To show this is 1 to 1, there's a couple of ways you can do it. You can just look at the graph and say it passes a horizontal line test. Or you can actually take the derivative of this up top here and say, Well, f prime is going to be what? The derivative that is just too, which is strictly larger than zero, which implies always increasing. And that is one of our, uh, types of functions that we know is 1 to 11 that is always increasing. So we have that now they want to graph f inverse. So we don't even need to know what FM verse is because remember when we're looking at this. So if we have, like some point X Y here on f inverse, this is going to get flipped to become y X, so we can just take a couple of these points and flip them. Or actually we can take all three of those points and just flip them. So, um, this point here is negative. 30 So then that's going to get flipped over and become negative three. Or this was zero negative. Three. We can flip that over. Become negative. 30 this here is negative. One or one negative one. So that is going to become negative. 11 Yeah. And then we can take this point here, which is 21 and that is going to become 12 to look something like this. And then they would probably intersect somewhere around here. And you could find where these intersect at. Um, I don't know why I said, why is he going to X here? Let me erase that. Um, this is why is into f of X is what I meant to write. And then this over here is why is equal to, um, that inverse of X and one way you could actually used to find where is this intersection point out is by using the fact that we know these should be reflections across the line. Why is equal to X? Um so, yeah, if you were to just, like, set these two equal to each other and then solve we can actually find with it. So, actually, let's just go ahead and do that to make it a little bit more accurate. So I'll do that off on the side. So X is equal to two X minus three. Um, so you add three or subtract so that gives us X is equal to three. So actually, it looks like I did a pretty good job there with that. Just luckily, um, yeah, so we know that is where these two are going to intersect. So now we have those graphs. Um, the next thing. So we did eat. Now it's time to find the range of domain. Well, remember, all we're going to do is look at our original function and then switch these around. So the domain of the inverse is the range of our original function. So the domain of F inverse is just the range here. So that's still negative. Infinity to infinity. And then the, um, range of FN versus just our domain, which is negative. Infinity to infinity. Okay, then let me actually skip this down a little bit for the next one. So if we want to find the inverse. Now we can go ahead and first change this out for why? And then switch the excess and the wise. So this is going to be X is equal to two y minus three. And then we just need to solve for y so we would add three over divide by two. So that would give us y is equal to X plus three all over two. And then this is our adverse. So that is the inverse. And then finally, if we want to show, um that both of these compositions just give us X. We can go ahead and do that. So I'll start with this first one over here. So we have f inverse of f of X. So this is the same thing as so we're just going to take f of X and plug it into f inverse. So the f of X plus three all over two, and then we can go ahead and plug in ffx, which was two x minus three. So the two x minus three plus three all over two. So those threes cancel out, Then we get to X over to the twos council, and we're just left with X. So this first one checks out. Now we can go ahead and do the same thing. But when we flip so f of f inverse of X, so now we're gonna plug F inverse into ffx. So this would be to F inverse F X minus three, and then f inverse is X plus 3/2 minus three. So those two's cancel out and then we have X plus three minus three threes. Cancel and then we're just left with X, so that one also checks out as well.

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