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Glve th dedntion 0l ech ol thn _olowng Give the dednition J :A being functionGiven 34 B i; function jve the dedlnition for bing jective)1in-Given (Surjection}fundio...

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Glve th dedntion 0l ech ol thn _olowng Give the dednition J :A being functionGiven 34 B i; function jve the dedlnition for bing jective)1in-Given (Surjection}fundiondcfinitionGIvCn function give the dednition for bcing conespondenoe (Bijection)Givedednition for P € being TNInnHicz

Glve th dedntion 0l ech ol thn _olowng Give the dednition J :A being function Given 34 B i; function jve the dedlnition for bing jective) 1in- Given (Surjection} fundion dcfinition GIvCn function give the dednition for bcing conespondenoe (Bijection) Give dednition for P € being TNInn Hicz



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Determine whether each of these functions is a bijection from $\mathbf{R}$ to $\mathbf{R}$ .
a) $f(x)=-3 x+4$
b) $f(x)=-3 x^{2}+7$
c) $f(x)=(x+1) /(x+2)$
d) $f(x)=x^{5}+1$

Hello strings in this question, Set A. And set B two sects are given and we have a function F is defined from A to B. And the definition is F X equal to access square. So let me draw mapping diagram here, set a contains -101 and two. While the said we contains 012 and F X is X square. It means the image of X is x square, so therefore minus one will be minus one is square, that is one. So I will relate minus one veteran F of zero is zero square, that is zero, so image of zero is zero, Therefore one is 1 square, that is one. So I have one very map with vitamin And f of two is 2 square, that is food. So I will relate to with four back since there is not for in the said be So I cannot relate to with any element and therefore I can say that the state that element to here has no image in the set B, because four is absent from the set B. Therefore, that this cannot be a function because in a function, each element of the set A must have a unique image, and since this element to has no image, therefore, it cannot be a function so forth, so I should be the correct answer.

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.

All right, so this function here, it's not inductive. Yeah. Right. Um We can do a counter example. We can choose X 1 to be a negative one Annex 2 to be one. So we have X one is not equal to X two. So I have two different inputs. Um I should expect different outputs. Okay, But outputs we have F of -1, which is two absolute value of negative one, which is just one. So we have Have 91 is two And this is the same thing as a positive one. I have um the same output for two different inputs. And so this is not objective. And so if it's not objective we could say that it can be budget for it to be my objective. It has to satisfy interactivity and subjectivity.

All right. So I want to see this is um my objective and so our graph of the exponential. I mean that's the value function looks like this. Okay. And as you can see that this is not either right objective and it's not its objective. Okay. Every output doesn't have and input if you do the horizontal line test and every output or end, there are some outputs that have more than one input. So for so because of this here it's not injected right? Every output has to have the most one input. And because of everything, right, uh all the negative real numbers um that do not have a corresponding input is not subjective, so it doesn't hold both of these conditions, it's not going to be budget.


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