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(16 nute) Fetht tuRt013171aennt (ie exblicitly coutTct function htom Imt Your funetlon i1 bijection)...

Question

(16 nute) Fetht tuRt013171aennt (ie exblicitly coutTct function htom Imt Your funetlon i1 bijection)

(16 nute) Fetht tuRt 013171 aennt (ie exblicitly coutTct function htom Imt Your funetlon i1 bijection)



Answers

If $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ is a bijection, then (a) $\mathrm{n}(\mathrm{A})=\mathrm{n}(\mathrm{B})$ (b) $\mathrm{n}(\mathrm{A})>\mathrm{n}(\mathrm{B})$ (c) $\mathrm{n}(\mathrm{A})<\mathrm{n}(\mathrm{B})$ (d) $\mathrm{n}(\mathrm{A})=\mathrm{n}(\mathrm{b})+1$

We're the function ever backs to find the Aztecs minus one, divided by two ex woman sex. So I've plotted the graph off the function as a function are and, uh, we're asked to show that f were restricted to the interval. 01 finds a projection between the intervals are one and the set of real numbers. And here we see from the picture that we just look at the branch of the function between zero and one. They're basically starts at zero all the way down from minus infinity, and he goes up going towards one all the way up to infinity. And the idea is that is strictly increasing. So he only touches every single point once and also it touches all the ball the wise because it starts from Monets and feeding goes toe up plus infinity. And so it must touch every point. So they said, Yeah, let's show that by first computing the derivative of X disease Well, the square of the dominator And then in the numerator we have the derivative of the numerator, the multiply the denominator minus the numerator, the multiplies that the river TV over the denominator here I've computed all the products and it turns out to me for X squared minus for X Plus two and we see these as two X minus one squared plus one divided by four square woman is X squared and then an observation here is that these derivative is always positive because in the numerator we have a square plus one. So that's definitely positive, strictly positive and in the denominator as well. We have a positive number which could be at most zero, which was we have, well, plus infinity. So we know that the river TV's always zero and in five we knew that from the picture ready because we see that the function is always increasing exact for the issues that zero i one where we have a discontinuity point. So now that we know that function is always increasing, we also observed that the function is continues on the intervals minus infinity to zero excluded, then on the interval, 0 to 1 without the extra mile points and from 12 plus infinity. And of course, these because F is defined as the ratio of polynomial sze, so it must be continues except for the points where the denominator vanishes, and in this case you seem merely that. Those points are zero and one because zero is a zero off tracks and one is a zero off woman sex. And also we see that from the teacher that where where we have the left legalizing talks, so in particular if we look at F as a function only from 0 to 1. So I used this symbol to say f restricted to the interval 01 while dysfunction he's continues and is strictly increasing because we completed the River TV's Stickley positive. And also, if you take the limit to zero from the right of dysfunction, we said that that's minus infinity, and we take the limit toe one from the left of F. That's also plus infinity. And therefore, as you observed, efforts ticket to 01 must be injected because it's strictly positive and subjective because it goes from modest in vain to plus infinity in a continuous way, so it must touch every single real point. So this function is objection, which is what we were asked, and now we're as to prove the same results so that the intervals you wanted the same car tonality as the set of real numbers, but isn't sure that Bernstein with him. So let's compute one direction where we see that well, the inclusion map from 0 to 1, I mean for a needle 012 are the sense. A number axe to itself is clearly injected and therefore the calamity of the intervals. Your one is lower equal than the car Garrity of our for the other direction. We observed them up from our 201 given by X goes to to develop a pie are dungeons are contingent of X is inductive, and we know that because you can just computed derivative. The devotee off the arc Tangent is one over one plus the square, so we definitely know that it's increasing. And also there's a factor of two divided by buying because, well, they're contingent by itself goes from Manus Behalf, too, plus my health. So by scaling by over pie, we know that the image of this function is in 01 Anyway, this function is injected because it's strictly positive and he goes into 01 and so the unity of our is lower equal is granted equal than the carnality zero away. Not is a mistake. These were supposed to be lower recall. Of course, there we go. Which means that by combining these two information by the shutter Ben Stein theory, we know that the community of 01 must be equal to the community of our

Okay, so in this question once and determine if these functions are biting his own loss. So for every F or X is equal to two explosive. So for this function, this is a projection. So if we just photograph, we can see why it's by dishing. We say that it is a 1 to 1 function because every X value will only give us one y value and every single by value. We can get everything Wide Valley just by taking on extra that this is yes, a projection to formally prove that's a projection boat. Let's take the transformation. X is equal to X minus one divided about you and then we'll see what happens. Um, where X can be some room number. What room number. And then we'll see what happens when we put distinct into our function. Thanks, man. 20 right. Shoot. It's been two time by X minus one over two. Just one. This is X minus one plus one. So this is just sex. Since X is your room number, it follows that this is a 1 to 1. Now be, uh, Alex is you two x squared plus y. So this is not This is not a projection which is pretty pretty clean because X squared X squared is greater than the eggs. They're ex great. Plus one is gonna regret going away to one. So your function is going to regret it. Then what you want? Um, well, bye. Objection. Because we're taking we're continuing by Jackson's from our to our Then we're missing the entire negative domain. So you just draw a graph out for this question. We only had two domain for wise. Greater than zeroing. But we don't have wives better than one. Sorry, but we don't have anything that has been lowered. So therefore, this is not a distraction. Now C f o X is equal to x cubed plus one. So this is much easier if you destroy the graft. First execute plus one will look something like that and you'll see once again that every Y valley can give us one every by valley. There's an X value to meet it on every X value is different. Yeah, so therefore, this is Yes, it is a protection to prove that you take the map. X maps, too. The cube off X minus one where X is aroma. So you function evaluated at she X months. One evening to the huge bridge off X minus one. You just won the chamber it and include cattle. Cancel each other out. You have X minus one plus one. So this is just X. I said, this is This is your room number from this. This is Bart Station now. The F o X is you two x squared plus one over X squared, plus two. So this is not a projection. So the reason why this is not a by injection or x squared plus one is rather than a week to one x squared plus two is gonna greater than or equal to two. So what is mainly that top number is always going to be a positive number. Well, the chip number, the bottom number is also gonna be a positive number. So a positive number, the body by a positive number is always gonna be a positive number. So then this whole This whole thing here is a party member. I mean, that's greater than or equal to zero. Well, once again, the problem with with this question is that we were taking fire victim from Artie are as a result you bought when you completely miss out on all the Y values below the one negative allergies. So therefore, this is not a project because it's always can be president.


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