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

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. We're given five answer choices, options A through D to the graph of a relation. And option E is none of the above. So we're gonna go through each of these options A through D one at a time, try to figure out if they are correct. What the question is asking is to do two things. Kind of the first is we need to identify if it's a function. OK. If these graphs represent functions. And the second is whether or not it has an inverse function. All right. So let's start by going through these and figuring out which ones are actually functions. Now recall if we're giving the graph of a relation and we want to figure out if it's a function, we want to perform the vertical line test. So we're gonna go through and we're gonna perform that vertical line test. OK. And we called it for the vertical line test. What it says is that you can draw a vertical line anywhere along your graph. OK. So I'm just drawing some sample vertical lines. Um and they're dotted dashed. And what it tells us is that as long as that vertical line roses are curved only once, then this is a function. OK. So in this case, that's true. OK. Every vertical line we draw will only cross the graph once. So this passes the vertical line test. So we'll write V L T and we write a checkmark. OK. So this one is a function I'm gonna short form function to F N just to save some space. Now moving to the next graph in part B now this looks like a quadratic. So we expect that this is going to be a function and we can do our vertical line test. Again, we draw our vertical lines. We if you were to draw it anywhere, that line only crosses the curve in one point. OK. So this is a function, it passes a vertical line test. OK? And remember how that relates to the definition of a function. OK? If we draw a vertical line and it crosses our graph only in one point. What that means is that for every X value there's only one corresponding Y value. So that's how the vertical line test relates to the definition. Moving to the next graph in part C. And we can draw our vertical line anywhere where this function exists. And we are gonna cross that curve at most once as we have one intersection point never more than one. So again, this passes the vertical line test, we will give it a green checkmark for that. It is a function. And finally, to part D, this looks like a quadratic that's been tilted onto its side if we draw a vertical line on this graph. So we've drawn it at X equals five. We can see that it crosses our curve or our relation in two points. OK. So for this particular X value of five, we have two corresponding Y values. So this is not a function, it does not pass the vertical line test. We're gonna give it a red X and it is not a function. OK. So option D is already out because this is not a function. Remember we're looking for the graph that is a function and has an inverse function. OK. So we've dealt with the function part of this. We've narrowed it down to ABC or maybe E if none of them work out. And now we're gonna look at the inverse function and for the inverse, we do something very similar. But in this case, we're gonna look at a horizontal line test instead of a vertical line test. So we're doing the exact same thing, but we want to draw horizontal lines. Now for part A, if we draw horizontal lines at higher Y value, say Y equals two or three, it looks OK. And it's only crossing in one point. However, if we were to draw a horizontal line at say uh Y equals 0.5. You can see that it's gonna cross that curve in two places. This means that this function does not have an inverse. It does not pass our horizontal line test because it crosses in two different points. And so we say that the horizontal line test gets a red X and there is no inverse here, there is no inverse function. So we can eliminate option A. That's not gonna be the correct answer. Moving to option B again, we're gonna perform our horizontal line test. We're drawing these horizontal lines in blue. And for any horizontal line we draw that's above kind of the minimum of this curve. We can see that again, it's going to be crossing in two points and our curve is going to intersect our horizontal line in two points and one on the left, one on the right. And so this also doesn't pass the horizontal line test. It gets a red X, it has no inverse. OK. So we can eliminate option B. And finally, we're gonna look at option C, we draw our horizontal lines and unfortunately, we see the exact same thing if we draw a horizontal line on this graph. A relation crosses that horizontal line in two points. OK. So the horizontal line test is failed. Option C does not have an inverse. And so this is not the correct answer. And so what we found is that A B and C were all functions. They passed the vertical line test but they didn't pass the horizontal line test, which meant that they didn't have an inverse. And for option D it failed the vertical line test. It wasn't a function at all. And so the correct answer here is gonna be option E none of the above. 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

Inverse Functions

Horizontal Line Test

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