5

(d) Let X {1,2,3,4} and Y = {5,6,7}. State whether a function f : X ~+ Y can be one-to-one or not without justificalion. Also, produce an example of a function f:x ...

Question

(d) Let X {1,2,3,4} and Y = {5,6,7}. State whether a function f : X ~+ Y can be one-to-one or not without justificalion. Also, produce an example of a function f:x ~Y Ihat is not onto. [6 marks ]

(d) Let X {1,2,3,4} and Y = {5,6,7}. State whether a function f : X ~+ Y can be one-to-one or not without justificalion. Also, produce an example of a function f:x ~Y Ihat is not onto. [6 marks ]



Answers

If a relation $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ is defined by $\mathrm{f}(\mathrm{x})=\mathrm{x}+2$, where $\mathrm{A}=\{-1,0,1\}$ and $\mathrm{B}=\{1,2,3\}$, then $\mathrm{f}$ is (1) only one-one function. (2) only onto function. (3) bijective. (4) None of these

In this question. The function provided to us. If I can say that we did a rational function that is attacks equals to three access where Last three x -4 and divided by three plus three x minus four. School Okay we have to check that it is 11 hour lot and it is on to or not. Okay let I take here at X is equal to zero. Then you can see here from here I got three x squared plus three x minus 40 because Toledo and you can see that this condition provides you to a loser effects. Okay? By using the formula that it's minus b plus minus square root, the square minus pleurisy means -4 into today into minus four is divided by two means two and 23 Okay from here we were the two gallons that is latex one next to and you can see that both are not take well but what is happening here is that X one and next door. Look at those values that are not equal. But you can see both of them have the same images and the immediate zero. Because we are sold by taking them zero. Okay, so we can say that the function is not 1-1 here. Okay, now we have to check that it is on to or not. So for that I'm going to find the range of the function. Okay, let's take the function. Why he requires to affects here means like the first two They're given function. That is I can say that three axes square Plus three x -4 and divide by three plus three X minus four excess work. Okay, nowhere to further solve it. So you can see that after cross multiplying is change it changes into quality equation that is access. We're into three plus four way And plus X into 3 -3 ways and- of 4 -3 way The Cholesterol zero. Okay now you see whatever your X. Must be, it should be always a real number. Okay because you cannot put here in misery numbers so it should be real number. And due to the region that actually real number, we have to apply the condition that the big square minus forest should be better than it was 20 Okay means discrimination should be better than it was. A little bit square means three minus three wives. Foley square minus four in two ways. Read this far away and sales minus four minus three Y. Should be greater than it was okay after simplifying it regards the equation that is 57 Last 74 way plus 57 Y square should we gotta than it was +20 Okay. My were divided by 57 year. So after that I get the question in the home a white square plus 74 by 57 Y. Plus one. But then it was too little. Okay now I make your complete square. So after that I got here, Y plus 37 x 57's Holy Square. And less of 1880, divide by 3 to 49 Better than it was 20 Okay now we see that this condition, he's saying that the value of this exploration should be good. Then it goes to zero and there is complete square here. So I can say that for I need the loo of why this in its ability satisfies here. Okay. You can see that if we put any value of why did the inequality is going to satisfy here? So we can say that. Why maybe any real number and why maybe a serial number And why is our range. So we can say that he arranges are. And you see in your question that is it was too cold to men because co domain is also are here. And due to this reason that ranging women are equal, you can say that the function is on to function here. Okay, So in this question, what we got that the function is not 11 Okay, but it is on to. So I can say that the result is not one, but on two. Okay. And this difficulty is given an option number B. You can see option number B says that the function is onto but not 1 to 1. Okay, so the option number B is correct. Answers for the given question here. Thank

In this question, it is told that there is a function defined formats to wait here and it is us that when your f in what exists here. Okay, so you see that let them talking about the 11 of this function here. Okay, this is their tax. This is 3rd way. And in this we have 12 and three. And in this we have a N B. Here. Okay, now letter I'm mapping like this way here. You can see that the function is not 11 here. Okay. And now, if we make the inwards here, okay. Means, from why it works for this situation, you can see here, we have to map with one and two and we have to map bay with treats. So you can see that this is your inverse function but it is not a function. Okay, so you can say that the function must be 11 here to find the universe. Okay, So the necessary condition is must be 11 here. Now we talk about onto. Okay, so again, I'm going to take the third accent y here. Ok. The attacks have one and 2 and why have a B&C? One is mapped with? The two is marked with. Okay, this was your functional. Now we make the universe of the situation here. Okay, why is having a B C axis having one and two? We have no map A with one and baby too. I can see that I found the universe. Okay. You see here, it is not On two. Okay. And you can see this is not a function here. Okay, so the basic condition here also that it must be onto also. Okay, so now again included F universe exists. Yes and only if interface 11 and on. Okay. This is the necessary condition for inwards to we exist here. Okay. And this is given the option. So, the best choice of this question here is I can say that the option. Okay, thank you.

Ah, function is 1 to 1. If there's no horizontal line that intersects the graph at more than one point now, looking at the image in the textbook, there is no wars on tow line that crosses the graph at more than one point. Therefore, it passes the wars on tow line test. So it is one toe one. So yes, it is 1 1 1 1 to 1.

All right, So on this problem, we are asked if this function is 1 to 1. I am going to use the horizontal line test and let me see if I can get this lined up here. So the horizontal line test. If I start here and I go across, you can see that only touches in one place. Let's see if I can go up a little. I go here and come across. It only touches in one place. Maybe if I go down a little thio here, it only touches in one place in down a little more one place, one place in one place. So no matter where I perform the horizontal line test and even though this might get a little sketchy through here, um and it might be hard to see because this function continues to increase its still. Even though it looks weird, it's still on. Lee touches that horizontally and one place. And it might be different on Lee by a decimal, but not not by much. So the answer two, this is Yes, it is 1 to 1


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