5

X. by - [Ty ` a bL € : (f} CLcyt ( 1 ? L^ DLO[cz? FJ , J #0]ns ] {iTR 01'4 Heanernon ["[ TucombTere KECEIAE LMk" #KOBFEW? 2h0n VTT KokK AOAcaAL...

Question

X. by - [Ty ` a bL € : (f} CLcyt ( 1 ? L^ DLO[cz? FJ , J #0]ns ] {iTR 01'4 Heanernon ["[ TucombTere KECEIAE LMk" #KOBFEW? 2h0n VTT KokK AOAcaAL {Flesse Frict)SHOW ALL WORK" ENSURE MAXIMUM CREDITJ1Find the inverseof the function below and show that f(f (x)) =f(x) = 20 - In(2x)>0

X. by - [Ty ` a bL € : (f} CLcyt ( 1 ? L^ DLO[cz? FJ , J #0]ns ] {iTR 01'4 Heanernon ["[ TucombTere KECEIAE LMk" #KOBFEW? 2h0n VTT KokK AO AcaAL {Flesse Frict) SHOW ALL WORK" ENSURE MAXIMUM CREDIT J1 Find the inverse of the function below and show that f(f (x)) = f(x) = 20 - In(2x) >0



Answers

Verifying Inverse Functions In Exercises $21-32$
verify that $f$ and $g$ are inverse functions (a) algebraically
and (b) graphically.
$$f(x)=\frac{1}{1+x}, \quad x \geq 0, \quad g(x)=\frac{1-x}{x}, \quad 0<x \leq 1$$

Hi, everybody. This is section 4.8, number 20. And the question tells us for the function F of X is equal to the square root of X. We have to find the inverse function and then also find the domain and range of the original function and its inverse. So first, let's finance first function, we're just gonna write it as y equals a squared of X. And then to find the inverse we switch the X and y. So now we're going to get X equals the square root of why we wanna square both sides to get rid of the square root. And we get why is equal to X where, which means our inverse function is f inverse of axe is equal to X squared. Now there's a little bit of a problem with this. We're gonna look in a minute. Let's actually look at the graph. First seconds understands it a little bit better. The original function will do that in black. That swearing backs that graph looks like this, right? You can have any negatives under the square root, so it starts at zero, but you can have all the other X values so for to make a little bit of a chart for the domain on the range. Let's make a little bit of a chart for F. I tried, OK, after back on and f inverse of X. Okay, here we go. Okay. So for F of X, we could see the domain right from left to right. It starts at zero, and then it's gonna go on forever. So it's gonna start at zero and then go on forever. So from zero to infinity, the range Well, the lowest to highest. It also starts at zero and is going up for efforts in the range also is from zero to infinity. Now, if we're gonna look at the inverse function, the inverse function is a reflection in this line. Why is equal to X y? So if we're to graft distal, just do us a little bit better. It's a little bit more like that. There you go. Okay, so we're gonna look at this graph. X squared is a parabola, right? And the parabola would look like this. The problem is, that whole graph is not reflected from the black one from the original function. Just the right heart is broke. Lunar one after. Let's try that. We want to just see the restriction, right? We have the original graph, and then we have the inverse. That will be a reflection over the line. Why will stacks. So the issue is we have to restrict the blue one right, which is our X squared, and we're gonna have to restrict it. That it goes from zero to infinity could only start here. We don't want the other part of the problem, so the inverse would be f inverse of X equals X squared, where X is greater than or equal to zero. Another way to realize that is well, this domain of the function has to be the range of the inverse because you switch ex ally values and the range of the function must be the domain of the range of the original function. Right? Must be the domain of the inverse. So it happens to all be the same in this case. But because you switched X and Y for the inverse function, you'll end up switching the domain on the range

So our next function is a problem. But notice that in this problem we don't have the entire problem. We have x squared minus 10 X. And we want that to be where x is greater than or equal to five. And notice if I graph this, let's just do a little analysis on this. If I factor out the X that graphically this problem has zeros at town and it's going to have a vertex at five and this graph is going to do this and we're only going to have this part at the function. So this will end up being a 1-1 function and are inverse, will be a square root function. And we could figure out what this vertex was plugging five back into here. Plus five in there and plug five in there. Right? And find out that's five times negative five. So negative 25. So we have Our domain is we can plug in numbers that are greater than or equal to five. And we're going to get why values that are greater than or equal to negative 25 and are inverse F in verse. We'll have the feature that will only be able to plug in numbers that are greater than or equal to negative 25 get numbers that are greater than or equal to five. So let's find our inverse. So we know we want to switch them the X and Y around. Let's do it in blue. So we have y squared minus 10. Y must equal X. Now we're gonna need to complete the square. So we know to complete the square to solve for Y. Here, we need to take half of this and square half that's negative five squared is 25. So we're going to add 25 to both sides Or we can add 25 and subtract 25. And so this becomes why -5 quantity squared is equal to X-plus 25. Now we're going to take and typically we would take the plus or minus the square root of the X plus 25. But we're going to only want the positive part because of the part of the problem that we had to begin with. So we really only need this plus and then we have why is equal to the square root of Absolutely Race that Little five. We have the x plus 25 Plus five. And notice that we can plug in numbers that are greater than or equal to negative 25. And the smallest this part can be is zero. So I always output numbers that are greater than or equal to five and that's what we needed for our inverse to have to happen. So there is our Denver's

Let's find the universe of following function after that equal X squared plus two. And we want to restrict our domain to where X is positive is greater than or equal to zero. This is our original function now what's on the in first? So here's the step by step. The first step we want to do. Is there police the event with simple why this is semantic, but it would be helpful later on. Why equals X squared? Plus, we're not gonna rewrite the domain each time we'll keep it on the top of the beach there. The next step is this is the real inverting part. Replaced every wine equation with an ax and every x of y. What this does graphically is slip our grass over the line. Y equals that we'll see that on our graph in a moment. But first, let's do it. Algebraic Lee. Now we have thanks equals Why squared Plus to now to get into classic functional taste You want wide by itself On one side we have why squared equal minus to now let's hear both sides. Why equal either the positive or negative? Screw this. So when he took the square root. It gives us even a positive or negative option. So we need to figure out from our domain whether we want our wise to be positive for a wife to be negative. So, looking at our domain, this tells us that our input, our exes, should all be positive. So that means in the inverse all of the wives to be positive. So let's choose. The positive sides represent are in for assumption. We have. Why he quote the pounds. This will just leave it out. I my to this is our final answer for our interception. Now let us go grass. Each of these functions does it, bones. Okay, so here have already graft is the black line is the X squared plus two with the domain restricted. So it only draws it when I found them and you get a nice practice and I drew the line y equals X. We can see the reflection Then first, let's drop without our out to break manipulation. Just were placed me comply with the next maniacs with a lot. Let's see what that looks like so that the perfect reflection over the line one just to double check our own work that our how to break rearrangement was correct. Let's do our graph of y equals square root of X minus to with you can't see it took a lot of for okay.

Hi, everybody. This is section 4.8. Number 18 on the question gives us the function G of X is equal to X minus five. We have to find the inverse of the function on dhe. Also describe the domain and range of both of them. So the first thing that we're going to do is find the inverse in order to find the inverse of a function. What we're doing is switching the X and y value. So I rewrote it. A little simpler G of X equals X minus five. So we have y equals X minus five. I we're going to switch the X and the why and now good X equals Y minus five. And now it's all for the new. I add five to both sides. And why equals X Plus five, which means our G inverse function is equal to X plus five. We have our inverse function and let's talk about the domain and the range for the original function. Our domain is going to bay. Well, really numbers. This is a line. There's no issues with the linear function. So we have the domain is going to be, well, riel numbers Paul rials. I don't realize, and then the range is going to be, well, really. Numbers also. Well, look at the inverse function Y equals MX plus be right. It's still in the form of a lot in a linear function. So again, every linear function will have the domain and range of all riel numbers double reels.


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