can a horizontal line represent a function?

y = 5 is a horizontal line and is indeed a function.

could he have just used the vertical line test?

Yes, he could've. If he did that, then he would've noticed that the relation intersects the vertical line x=-2 at infinitely many points. This is because the relation is x=-2, so obviously it intersects it at infinitely many points. However, I think Sal was trying to demonstrate a more rigorous way of testing a relation for being a function. Instead of just doing a vague, vertical line test, he used the definition of a function to test the relation for being a function. I hope this helps!

Can there be many domain but getting only one range? If the line will be horizontal will it be a function?

One domain and one range although the domain can consist of the union of various regions on the x-axis. EG {-100 < x < -10} U {-1 < x < 1} U {10 < x < 100}. _the U is the symbol of union_. A horizontal line is a function, but a pretty boring one since no matter what x value you input, the output will always be the same. EG f(x)=5. No matter what x is, the output is always 5. As you can see, the output value does not depend on the input value x.

So is Sal saying that x -> f(x) -> infinity is not a function? If you just wrote the infinity sign could it be considered as only one output?

Infinity cannot be a single output. This rhetorical question I'm about to give you came from another user: "Think of the biggest, biggest, biggest number you can then keep adding 1." There is no definite answer for infinity, so it can't be considered as a single output.

is a function with multiple outputs a logarithm?

No. A function, by definition, can not have multiple outs for a specific input value. Each input can create only one output to be a function. Thus, any equation that doesn't meet this definition would not be a function. FYI.. there are logarithmic functions.

Would a drastically curved line on a graph represent a function? Does the curve have to go through x and y to be a function?

For a relation to be a function, use the Vertical Line Test: Draw a vertical line anywhere on the graph, and if it never hits the graph more than once, it is a function. If your vertical line hits twice or more, it's not a function. For example, a circle is not a function because when you draw a vertical line on top of its graph, the vertical line will cut through the circle twice.

What's the vertical and horizontal line test?

The vertical line test is used to determine if a graph of a relationship is a function or not. if you can draw any vertical line that intersects more than one point on the relationship, then it is not a function. This is based on the fact that a vertical line is a constant value of x, so if there is one input, x, with more than two outputs, y, then it breaks the function rule. A horizontal line test does not have as much meaning.

Main content

Course: Algebra (all content) > Unit 7

Lesson 7: Recognizing functions

Does a vertical line represent a function?

Google Classroom

Explaining why a vertical line doesn't represent a function. Created by Sal Khan.

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Ronit Patel
Posted 9 years ago. Direct link to Ronit Patel's post “can a horizontal line rep...”
can a horizontal line represent a function?
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(19 votes)
Answer
- Thomas B
  Posted 9 years ago. Direct link to Thomas B's post “y = 5 is a horizontal lin...”
  y = 5 is a horizontal line and is indeed a function.
  Comment on Thomas B's post “y = 5 is a horizontal lin...”
  (23 votes)
Teddy Karrer
Posted 11 years ago. Direct link to Teddy Karrer's post “could he have just used t...”
could he have just used the vertical line test?
Button navigates to signup pageComment on Teddy Karrer's post “could he have just used t...”
(11 votes)
Answer
- Noble Mushtak
  Posted 11 years ago. Direct link to Noble Mushtak's post “Yes, he could've. If he d...”
  Yes, he could've. If he did that, then he would've noticed that the relation intersects the vertical line x=-2 at infinitely many points. This is because the relation is x=-2, so obviously it intersects it at infinitely many points.
  
  However, I think Sal was trying to demonstrate a more rigorous way of testing a relation for being a function. Instead of just doing a vague, vertical line test, he used the definition of a function to test the relation for being a function.
  
  I hope this helps!
  Button navigates to signup page
  (41 votes)
ursaciucantonio
Posted a year ago. Direct link to ursaciucantonio's post “3.14159, he actually star...”
3.14159, he actually started defining pi :)
Button navigates to signup pageComment on ursaciucantonio's post “3.14159, he actually star...”
(9 votes)
Answer
- Taders09
  Posted a year ago. Direct link to Taders09's post “yes I saw that to. :)”
  yes I saw that to. :)
  Button navigates to signup page
  (6 votes)
areesha.kadiwal
Posted 8 years ago. Direct link to areesha.kadiwal's post “Can there be many domain ...”
Can there be many domain but getting only one range? If the line will be horizontal will it be a function?
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(4 votes)
Answer
- Stefen
  Posted 8 years ago. Direct link to Stefen's post “One domain and one range ...”
  One domain and one range although the domain can consist of the union of various regions on the x-axis. EG {-100 < x < -10} U {-1 < x < 1} U {10 < x < 100}. the U is the symbol of union.
  A horizontal line is a function, but a pretty boring one since no matter what x value you input, the output will always be the same. EG f(x)=5. No matter what x is, the output is always 5. As you can see, the output value does not depend on the input value x.
  Comment on Stefen's post “One domain and one range ...”
  (2 votes)
Amy D
Posted 7 years ago. Direct link to Amy D's post “So is Sal saying that x -...”
So is Sal saying that x -> f(x) -> infinity is not a function? If you just wrote the infinity sign could it be considered as only one output?
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(2 votes)
Answer
- Johnathan
  Posted 7 years ago. Direct link to Johnathan's post “Infinity cannot be a sing...”
  Infinity cannot be a single output. This rhetorical question I'm about to give you came from another user: "Think of the biggest, biggest, biggest number you can then keep adding 1." There is no definite answer for infinity, so it can't be considered as a single output.
  Button navigates to signup page
  (6 votes)
Haya Aboudan
Posted 7 years ago. Direct link to Haya Aboudan's post “is a function with multip...”
is a function with multiple outputs a logarithm?
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(2 votes)
Answer
- Kim Seidel
  Posted 7 years ago. Direct link to Kim Seidel's post “No. A function, by defin...”
  No. A function, by definition, can not have multiple outs for a specific input value. Each input can create only one output to be a function. Thus, any equation that doesn't meet this definition would not be a function.
  
  FYI.. there are logarithmic functions.
  Comment on Kim Seidel's post “No. A function, by defin...”
  (4 votes)
Rusty Jr
Posted 3 months ago. Direct link to Rusty Jr's post “whats the vertical line t...”
whats the vertical line test
Button navigates to signup pageComment on Rusty Jr's post “whats the vertical line t...”
(3 votes)
Answer
ch3wbacc4
Posted 9 years ago. Direct link to ch3wbacc4's post “so in this graph, y is no...”
so in this graph, y is not a function of x but x is a function of y?
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(3 votes)
Answer
- BlondSpaceMonkey
  Posted 12 hours ago. Direct link to BlondSpaceMonkey's post “Saying that "`y is a func...”
  Saying that "y is a function of x" means that x is the input and y is the output, which is the system that most people use.
  
  If you were making a function and you decided to label the input "y" and the output "x" then you could say that "x is a function of y." But only if you did it that way.
  
  Since Khan Academy and a lot of other mathematicians all do it this one way, it makes sense to do the same.
  
  TL;DR: No, x is not a function of y.
  Button navigates to signup page
  (1 vote)
Johnny
Posted 6 years ago. Direct link to Johnny's post “Would a drastically curve...”
Would a drastically curved line on a graph represent a function? Does the curve have to go through x and y to be a function?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- RJacob1
  Posted 6 years ago. Direct link to RJacob1's post “For a relation to be a fu...”
  For a relation to be a function, use the Vertical Line Test: Draw a vertical line anywhere on the graph, and if it never hits the graph more than once, it is a function. If your vertical line hits twice or more, it's not a function. For example, a circle is not a function because when you draw a vertical line on top of its graph, the vertical line will cut through the circle twice.
  Comment on RJacob1's post “For a relation to be a fu...”
  (5 votes)
Sam D
Posted 7 years ago. Direct link to Sam D's post “What's the vertical and h...”
What's the vertical and horizontal line test?
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(1 vote)
Answer
- David Severin
  Posted 7 years ago. Direct link to David Severin's post “The vertical line test is...”
  The vertical line test is used to determine if a graph of a relationship is a function or not. if you can draw any vertical line that intersects more than one point on the relationship, then it is not a function. This is based on the fact that a vertical line is a constant value of x, so if there is one input, x, with more than two outputs, y, then it breaks the function rule. A horizontal line test does not have as much meaning.
  Comment on David Severin's post “The vertical line test is...”
  (4 votes)

Video transcript

In the following graph, is y a function of x? So in order for y to be a function of x, for any x that you input into the function, any x for which the function is defined. So let's say we have y is equal to f of x. So we have our little function machine. It should spit out exactly one value of y. If it spits out multiple values of y, we don't know what f of x is going to be equal to. It could be equal to any of those possible values for y. So let's see if, for this graph, whether for a given x it spits out exactly one y. Well, the function seems to be only defined so the domain of this function is x is equal to negative 2. That's the only place where we have a definition for it. And if we try to input negative 2 into this little black box, what do we get? Do we get exactly one thing? No. If we put in negative 2 here, we could get anything. The point negative 2, 9 is on this relation. Negative 2, 8 is on this relation. Negative 2, 7; negative 2, 7.5; negative 2, 3.14159-- they're all on these. So if you put a negative 2 into this relation, essentially, you actually get an infinite set of values. It could be 9. It could be 3.14. It could be 8. It could be negative 8. You get an infinite number of results. So since it does not map to exactly one output of this function, in the following graph, y is not a function of x.