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### Course: Algebra 1>Unit 8

Lesson 7: Recognizing functions

# Does a vertical line represent a function?

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

## Want to join the conversation?

• 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.
• can't you just do the vertical line test
• Absolutely, it's the simplest way.
• 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.
• whats the vertical line test
• What about a horizontal line? I imagine it would be an infinite number of inputs each with the same output. I'm not sure if that would be a function.
• You assessment is correct. A horizontal line has inputs of all real numbers and its output is always the same number. It is a function because each input creates only one output. The rule for functions does not care that multiple inputs happen to create the same output.
• of course pi is the first number that comes into sals head...
• so in this graph, y is not a function of x but x is a function of y?
• 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.