Which graphs in Exercises 29–34 represent functions that have inv...

Question

Which graphs in Exercises 29–34 represent functions that have inverse functions?

Accepted Answer

Hey, everyone in this problem for the following graphs, we're asked to identify which one is represented by a function with an inverse function. OK. So we're given four graphs here, four answer choices. And we need to determine which one is a function that has an inverse. So the first step is gonna be figuring out which one of these is a function. OK? See if we can eliminate any base on that and then we'll move on to whether it has an inverse. So for option A when we're looking at whether or not a graph is a function, OK? Whether the relation that's graphed is a function, we want to be thinking about the vertical line test. OK. So we want to think about drawing a vertical line anywhere along our graph. I'm gonna put four kind of sample vertical lines. But this has to be able to pass at every single point at every single vertical line that you could draw. And we want to see whether that vertical line crosses our graph in only one place. OK. So there's only one intersection between our relation and there's a vertical line. And in this case, you can see that there is OK. For each vertical line, it only crosses in one place. And that tells us that for any X value that we have, there's only one corresponding Y value in which by definition means that this is a function. So for option A, it does pass the vertical line test. So we're gonna write V L T for vertical line test and we're gonna give that a check mark. OK? And what that means is that this is a function. Now we're gonna move to B and in the graph for B, if we draw a vertical line, say that X equals negative three or negative four around there, you'll notice is that this crosses our relation in two points. OK? So our vertical line crosses our relation in two points. So for this X value which is around X equals negative four, we have two corresponding Y values. OK? And that's true for almost all of the X values. And so this does not pass the vertical line test and this is not a function. So we're gonna write V L T, we're gonna give this a red X. OK? This does not pass and it is not a function. OK? So if it's not a function, then it cannot be the correct answer. So we can eliminate option B moving to option C, you can go ahead and do a vertical line test again. We see that for any vertical line we draw along this curve, it is only going to intersect our relation in one point that means it passes over to Poli test and it is a function. And again, we're looking at these intersection points. So one intersection 10.1, intersection 0.1 intersection point for each of those vertical lines versus in B where we had two, moving to option D drawing our vertical line in this case at X equals negative five. But you could choose to do it at any X value that vertical line is going to cross our relation or intersect in two points. OK? In the positive Y axis and also in the negative Y axis. And so for every X value there is more than one corresponding Y value. So this does not pass the vertical line test and it is not a function. So again, just like for option B, we can eliminate option D since it's not a function, now we're gonna move on to the second part of the problem. We want this to be a function with an inverse function. So for an order, sorry, in order to have it have an inverse function, what we wanna do is look at our horizontal line test, OK? And we only need to do that for option A and C because we've already eliminated option B and D, our horizontal line test very similar to the vertical line test. We wanna be able to draw a horizontal line test at any point and have it only cross our relation in one spot. However, for option A, you can see if the, if we draw a horizontal line kind of around Y equals four, we are getting two intersection points. OK? So option A does not pass a horizontal line test that tells us that it does not have an inverse. OK. So for option A, there is no inverse. So option A cannot be the correct answer. So at this point, we've eliminated A B and D, we expect that C is going to be the correct answer. But we need to do our horizontal line test there as well just to double check. And if we do our horizontal line test, we draw our horizontal line anywhere along our curve and we can see that there will always be only one intersection point. OK. For any horizontal line we draw with our relation. What that tells us is that it passes the horizontal line test and it therefore has an inverse. So option B and D did not pass the vertical line test. That means that they are not functions. So we eliminated those option A was a function passed the vertical line test but it did not pass the horizontal line test. So it did not have an inverse. But option C passed both the vertical line test and the horizontal line test meaning it was a function with an inverse. And so the correct answer here is option c thanks everyone for watching. I hope this video helped see you in the next one.

Which graphs in Exercises 29–34 represent functions that have inverse functions?

Key Concepts

Definition of a Function

One-to-One Functions

Horizontal Line Test

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