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5 (0.4) Let X = {1,2,3,4,5} and Y = {1,2,3}. Classify the type of following functions as injective; surjective, bijective, Or none of these. Make sure_YOU explain_w...

Question

5 (0.4) Let X = {1,2,3,4,5} and Y = {1,2,3}. Classify the type of following functions as injective; surjective, bijective, Or none of these. Make sure_YOU explain_why:2 3 4 a) f:X - Y defined by f 5) 2 3 2 b) f:Y _ X defined by 1) X if x is even f:X -Xdefined by f(x) = X+1 if x is odd

5 (0.4) Let X = {1,2,3,4,5} and Y = {1,2,3}. Classify the type of following functions as injective; surjective, bijective, Or none of these. Make sure_YOU explain_why: 2 3 4 a) f:X - Y defined by f 5) 2 3 2 b) f:Y _ X defined by 1) X if x is even f:X -Xdefined by f(x) = X+1 if x is odd



Answers

Let $\mathrm{f}(\mathrm{x})=5 \mathrm{x}^{2}+3 \mathrm{x}-4, \mathrm{x} \in \mathrm{R}$. Then $\mathrm{f}(\mathrm{x})$ is a/an (a) periodic function (b) odd function (c) even function (d) neither even nor odd

So for these questions, we have to determine if these functions are by objections from room to room numbers. So it's prevented by objection. We have to show that it is a 1 to 1 function and is on to function. So once you want means that if F as a function off ex isn't t o f as a functional why and this implies that X is equal to why is that? This is 1 to 1, 21 And if it's onto, then we need that for every element in your if your function maps or really real numbers to roll numbers me that for every elements in your room number, your number there exists and elements in the room number such that as a function off X is big too. Why, so the first case, eh? Yes, as a function off X, is it two minus three X plus four? Okay, so this is a by objection. So this is a projection. So first proving that's a 1 to 1. So you have f a ce function off a Is it f as a function off B, you have minus three a plus for victim minus three D, That's four So minus three A Is it a minus three? Be so a is eager to be That's a once one function. Now we need to show that Let's let why be in your rooms so to generate Why all you need to do is need to say as a function all full minus Why divided by three. So you execute your minus three times like all my wide Dubai free plus four. So the 23 chances out So this is minus four plus why possible? Which is just keep Why so yes, this is a projection. Now she oh f as a function off X is equal to minus three x spread plus seven Okay, less do who by Objection. So if you have yeah, as a function off a Is it a za function of B? So you have minus three. A squared plus seven is equal to minus three. Be sweat plus seven. So then you have minus three a square is exit Mars three be sweat. A square is a beast with so a zing to plus or minus B So this is much a once 1121 isn't I wanted one. And so therefore it's my objection. A lot of objection. No, f as a function off X is equal to X plus one over X plus two. So this is more of a projection, not a projection, because there is no number X Such actor value F as a function of X is equal to one. So because this is a case is not a projection. Yeah. Now, finally, it has a function. All X is equal to X to five plus one. So yes, this is a projection as it so proving a function of A's a za function of B So a to the fight war, which implies a five plus one, is even to be the five plus one a five piece even to be to the fine. So a is equal to be Now, if you that's why be in your room notice then you all you need to do is say the function off the fifth ribs. Oh, why? Minus fool. And so I give you, uh, this route off Wine wars 15 power plus one. This is why must one plus one, which is why so, therefore, yes is a projection

In this question. The given function is hey folks access, it was to access word minus one into Access were -2 here. Okay. First of all we can see that in this case can be written as x plus one and two X -1 into access with -2. Okay. And their domain is are here. You can see that means the function is defined for all real values. We stay here. If we put here minus one then this provides at minus one plus 10 and if we put the other one then the second fun factor converts into zero. So you can see the minus one and one in order equal here. But you can see value it. Both of them is equal due to this reason. We can say that what is happening here is that Excellent and X two belongs to the domain and are such that they are not equal. But you found the values equal for both of them means we can say here that the function is not 11 or I can also say that it is not inject you here. Okay, sort of the given options. The B option is corrected because it is saying that the function is not injected here. Okay, this is the answer here. Thank you.

In discussion. We are provided with the function. It is told that f of X as it was to two weeks. Bless one and D O F actually liquids too Three X plus two is there And we have to find here dX OK, so let's start here. We are provided with Geo affects it shows that G at fx is three X plus two. Now substitute here effects that it's two X plus one. So dear The works plus one is 30x. Let's too okay, know what I have to do is that latter attacks my work last one by any other? Very well let us stay here Now if we are taking less one SD so you can see that X should be replaced here by the minus one divided by Okay, make this substitution here. So you should get GFT that was 23 and two t minus one by two plus two. Okay, now we further solid so we should get the value of duty that is three D by two and minus three by two and blessed to hear. Okay so finally I got here GFK that goes to Three day by a two and -3 by do less. So it can be written as I can say that plus of one by two. Now we see here you've got the value of duty and you have to find the value of G X. Now the place A by X. So we got here simply G X. As it was to three x by two less. 1 x two. Okay this is the final answer and you can see that these are the answer is In your option number 1st. So I can say that option number three in your option number second so I can say the adoption number second is the answer for this given question here. Okay thank you.

In this problem of relation function in each of the following cases we have to he stayed with the function is 11 or onto or by objective. So first we have given that message. The function is defined from our to our And given by this age, fx is equal to three minus four X. So now first we will check for 11 So this is a check for 11 A function is said to be 11 function if F of X one is equals two. F of X two implies that excellent sort of B equals two extra. And now we are taking the values. See F of X one equals three minus four X one. And here the value are three minus four X two from here we conclude that this is possible when disturbed michigan's a lot and minus four X one is equals two minus four X two as a possible when extra energy close to actual. So therefore the function is 11 here's 11 and now we have to check for onto concerns. So again here we can write this as why is equal to fx and this is equal to three minus four X. And now here we have, Y is equal to three minus four X. So here Y is equal to three minus four X. Now we have to solve for X. So minus four, X is equal to white minus three. From here. We conclude that here X is equal to three minus Y divided with food. And now we have to put here this value in this problem. So this would be F X is equal to three minus four and here is the value of X. That is three minus Y. They've had with four and now we have to solve it. So 4 to 4, can we cancel out? And here we left three minus three plus white from here we say that F X is equal to white from here. We say that affects city goes to one. So that's why the funds and F is on two. So defense and F is on to consent. So this is 11 and onto or because we can see that difference difference in age by objective function. If offense an age when, when and onto. We see the difference and its objective and now second part. So second part says that you're different than if is defined from Earth to us. And this is defined by it. F X is equal to one plus access square. And now we have to check for 11 and onto. So if uh here this will be checked for one winner. So check for 11 So a fencer is said to be 11 If he had F of Excellence is equals two F of X two. And this implies that excellence should be equals two X two. So now we are taking Excellent. So Excellent would be one plus X 20 square. And here one plus extra square and also excellent. An extra belongs from the real number. So from here we conclude that Excellent square. It equals two extra square since Excellent and extra belongs from real number. Excellent. And extra belongs from the given real number. So here X would have to values Excellent would have to values and here also extra would have to values with changing sign plus and miners. So that's why we can say that here. Excellent is not equals two X two. So that's why F is not 11 So that's why the answer is not 11 And now we have to check for on two fronts. So again, here a function is said to be on two here and this will be why is equal to fx And here this would be one plus extra square. So also we have here we can say that why is equals two effects equals to one plus X squared from here we say that why is equal to one plus X square From here we conclude that excess where is equals two y minus seven. And from here we concluded exit equals two plus and minus and the road by minus one. And from here we say that this is equals two plus and minus. Underwrote one minus Y. And now when you put the value say ffx so this would be one plus the value of X is here under the root of one minus y. Holy square from here. When we solve it, this would be one plus. Now this has been revoked. So square they revoked. So one minus Y. From here we say that this is two minus wife so F. X is equals two to minus Y. But we want fx equals to why. So that's why. So that's why the function is not. So this would be not equals to way. So that's why we can say that the function is not onto, not onto. So therefore we say that difference and it went went but not onto not onto and not this is not one word and this is function is not onto also.


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