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## 8th grade

### Course: 8th grade > Unit 3

Lesson 12: Recognizing functions- 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 table
- Checking if an equation represents a function
- Does a vertical line represent a function?
- Recognize functions from graphs
- Recognizing functions from verbal description
- Recognizing functions from verbal description word problem

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

- could he have just used the vertical line test?(10 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!(38 votes)

- 3.14159, he actually started defining pi :)(5 votes)
- yes I saw that to. :)(3 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)

- 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)

- 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?(1 vote)
- 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.(5 votes)

- What's the vertical and horizontal line test?(1 vote)
- 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.(4 votes)

- Can a vertical line be represented by an equation?(2 votes)
- Yes. Points on a vertical line would all have the same X value, but could take on any value for Y. So there would be no Y value in the equation.

For instance, if the vertical line crossed the X-axis at x=2, then x=2 would be its equation!(2 votes)

- does anyone know how to solve this? :

"solve by graphing and check" y= -2x-2 and y=-4 ??(2 votes)**Graph to solve****: y = -2x -2 and y = -4**

y = -2x -2

Is in, which means we can look at the equation and see**Slope Intercept Form****the slope of the line is -2**, (because*it's the coefficient of x*), the second term is the**y intercept and also -2**, (*where the line crosses the y-axis*).

★**y-intercept**of -2, means…**when x is zero****, y is -2**.

So we're able to.**graph that Point**

Point at: (**0, -2**) ←y intercept

★ Since the**Slope is Negative**, -2, we know the line is**decreasing**, meaning it goes ↘️**downward from left to right**, and…

•A**Slope of -2**, means every time we move**one unit Right,**.*we move two Down*

★So let's**do that from the y intercept**!

If we're at (**0, -2**) and need to…**move one**,*to the Right*

that's*plus one*on x axis…**x = 0 + 1 = 1**

and for y we need,**two Down**

that's minus 2 on y axis…**y = -2 -2 = -4**

Now**we have another coordinate**on the same line!

Point at: (**1, -4**)

★(**0, -2**) and (**1, -4**)**Draw a line through both coordinate Points**, to finish graphing the first equation.

• Now to graph…**y = -4**

Notice there is no written slope in this equation, just a y intercept, this is because it's a…

↔️**a Horizontal Line, slope = 0**

It runs flat crossing y-axis at -4.

★**Find -4 on y-axis, draw a straight, side to side, ↔️ Horizontal Line through it**.

★**Look to see where these two lines cross**.

That (x, y) coordinate is the**Point of Intersection**between these two lines,*the only*x and y coordinate pair that make*both*of these equations True Statements.**Continued in Comments**(2 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.