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### Course: 8th grade (Eureka Math/EngageNY) > Unit 5

Lesson 1: Topic A: Functions- What is a function?
- Worked example: Evaluating functions from equation
- Worked example: Evaluating functions from graph
- Evaluate functions
- Evaluate functions from their graph
- Equations vs. functions
- Manipulating formulas: temperature
- Function rules from equations
- Testing if a relationship is a function
- Relations and functions
- Recognizing functions from graph
- Checking if a table represents a function
- Recognize functions from tables
- Recognizing functions from verbal description
- Recognizing functions from table
- Recognizing functions from verbal description word problems
- Checking if an equation represents a function
- Does a vertical line represent a function?
- Recognize functions from graphs

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# Does a vertical line represent a function?

Explaining why a vertical line

**represent a function. Created by Sal Khan.***doesn't*## Want to join the conversation?

- can a horizontal line represent a function?(20 votes)
- y = 5 is a horizontal line and is indeed a function.(24 votes)

- could he have just used the vertical line test?(11 votes)
- 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!(42 votes)

- Can there be many domain but getting only one range? If the line will be horizontal will it be a function?(4 votes)
- 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.(2 votes)

- can't you just do the vertical line test(4 votes)
- Absolutely, it's the simplest way.(2 votes)

- 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?(2 votes)
- 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.(6 votes)

- is a function with multiple outputs a logarithm?(2 votes)
- 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.(4 votes)

- whats the vertical line test(4 votes)
- 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.(2 votes)
- 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.(5 votes)

- of course pi is the first number that comes into sals head...(4 votes)
- so in this graph, y is not a function of x but x is a function of y?(3 votes)
- 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.(1 vote)

## 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.