Question
Functions & Non-functions: Graphically
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00:0003:08
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00:02: Let's quickly review how you can distinguish a function from a relation that is not a function, and do it just graphically looking at the pictures. So remember the definition of a function, and it's written down here, f is a function, if and only if, for any x there is exactly one y, such that f of x equals a y. So what that means is that if we choose an x that is in our function, that's in the domain of our function, that there can only be one y that that x is paired with. So, taking a look here at the first function, we see that we have our x-axis and our y-axis in the usual place. And if you choose an x value, say, this one right here, the corresponding y value is right about here, on the y-axis, and there's no ambiguity at all. I mean, whatever x value, this is the only corresponding y value is going to be here. And that's gonna be the case for any point you choose on the function, any value you choose on the x-axis. If you pick this value, there's gonna be one y value, it's gonna be somewhere way down here. If you pick this x value, there will be just one y value, which will be right here. 01:25: And you may recall that we call... When we're doing this graphically, we refer to this sometimes as the vertical line test. So, what that means is that any time you draw a vertical line on your graph, you can only intersect the actual curve one time. So you can see here that no matter where I draw a vertical line, I'm only going to be able to come in contact with this curve in exactly one place. Now, looking at this second curve, which looks familiar, probably a circle, it doesn't look as if we're going to pass this vertical line test, because if I draw that vertical line there, I find that I have actually been able to touch the curve in two separate places. 02:15: And you see what that means is that I'm gonna fail to meet this definition, because wherever I cross the x-axis, let's just call that x, it means that there's a y up here, and we can even just go and call that y1, and somewhere down here, there's another y, which we'll call y2, and the problem is that this x is matched with both of these Ys, and they're not identical. I mean, obviously, they're at different places, one of them's up here, positive, one of them is down here, negative. So I'm now failing this, because I now have found an x where there's not exactly one. In fact, I found two Ys in this case. So this relation here is still a relation, 'cause it's a set of ordered pairs, but it's going to fail to be a function, because it fails the vertical line test.
Close
Question 1
In a function, each x-value has ________ y-value.
Responses
A at least oneat least one
B exactly oneexactly one
Question 2
True or False: The first example is a function.
Responses
A TrueTrue
B FalseFalse
Question 3
True or False: The second example is a function.
Responses
A TrueTrue
B FalseFalse
Video Player
00:0003:08
None
Captions
Show Transcript
00:02: Let's quickly review how you can distinguish a function from a relation that is not a function, and do it just graphically looking at the pictures. So remember the definition of a function, and it's written down here, f is a function, if and only if, for any x there is exactly one y, such that f of x equals a y. So what that means is that if we choose an x that is in our function, that's in the domain of our function, that there can only be one y that that x is paired with. So, taking a look here at the first function, we see that we have our x-axis and our y-axis in the usual place. And if you choose an x value, say, this one right here, the corresponding y value is right about here, on the y-axis, and there's no ambiguity at all. I mean, whatever x value, this is the only corresponding y value is going to be here. And that's gonna be the case for any point you choose on the function, any value you choose on the x-axis. If you pick this value, there's gonna be one y value, it's gonna be somewhere way down here. If you pick this x value, there will be just one y value, which will be right here. 01:25: And you may recall that we call... When we're doing this graphically, we refer to this sometimes as the vertical line test. So, what that means is that any time you draw a vertical line on your graph, you can only intersect the actual curve one time. So you can see here that no matter where I draw a vertical line, I'm only going to be able to come in contact with this curve in exactly one place. Now, looking at this second curve, which looks familiar, probably a circle, it doesn't look as if we're going to pass this vertical line test, because if I draw that vertical line there, I find that I have actually been able to touch the curve in two separate places. 02:15: And you see what that means is that I'm gonna fail to meet this definition, because wherever I cross the x-axis, let's just call that x, it means that there's a y up here, and we can even just go and call that y1, and somewhere down here, there's another y, which we'll call y2, and the problem is that this x is matched with both of these Ys, and they're not identical. I mean, obviously, they're at different places, one of them's up here, positive, one of them is down here, negative. So I'm now failing this, because I now have found an x where there's not exactly one. In fact, I found two Ys in this case. So this relation here is still a relation, 'cause it's a set of ordered pairs, but it's going to fail to be a function, because it fails the vertical line test.
Close
Question 1
In a function, each x-value has ________ y-value.
Responses
A at least oneat least one
B exactly oneexactly one
Question 2
True or False: The first example is a function.
Responses
A TrueTrue
B FalseFalse
Question 3
True or False: The second example is a function.
Responses
A TrueTrue
B FalseFalse
Answers
GPT-4o mini
Here are the answers to the questions based on the information provided in the video transcript:
**Question 1:** In a function, each x-value has ________ y-value.
- **Response:** B exactly one
**Question 2:** True or False: The first example is a function.
- **Response:** A True
**Question 3:** True or False: The second example is a function.
- **Response:** B False
**Question 1:** In a function, each x-value has ________ y-value.
- **Response:** B exactly one
**Question 2:** True or False: The first example is a function.
- **Response:** A True
**Question 3:** True or False: The second example is a function.
- **Response:** B False