Main content

## 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

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Checking if an equation represents a function

CCSS.Math:

Sal determines if y is a function of x from looking at an equation. Created by Sal Khan.

## Want to join the conversation?

- Is it safe to say that if there are exponents in the relationship that it will not be a function? Is this generally true? Thanks(4 votes)
- Not necessarily.

Take the relationship y = x^2

y can be a function of x because every x value has only one y value.

But x could not be a function of y, because each positive y has two x values.(13 votes)

- but how did he make that arc? How will you know what shape to draw on a line?(2 votes)
- Not sure yet dude, but one thing I've found on Khan Academy is to trust Sal and then later on you find the answers down the track. I've found that when he introduces concepts you don't understand, generally it seems you don't need to understand how to do them at this stage (if you are following through a logical progression). Looking at an equation and being able to draw it is probably something that comes from a ton of experience!(9 votes)

- If you restricted the range to y>0 could you turn this into a function?(2 votes)
- Yes, because each x≥3 would be mapped to only one real number, namely the positive square root of x-3.(3 votes)

- So from what I understand a functions can't be any value that's taken to an even power (but the rules and inputs can be). What other mathematical concepts are there that prevent a relation from being a function?(1 vote)
- Couldn't I just plug this equation into my graphing calculator since they are allowed in most high school math classes and the math sats?(0 votes)
- That depends on whether you understand the concepts. It is fine to have software to help you get through the tedious computations, but if you do not understand how to do the graph yourself, your calculator won't be much help.

For example, you may be given a graph and asked what equation it represents. Your graphing calculator won't help you much there.(7 votes)

- i have a friend who does not understand functions. he can't wrap his brain around it. what tips can help him understand?(2 votes)
- A function is a relationship or expression relation one ( or more ) inputs to an output.

You are just relating two numbers with a certain rule.(2 votes)

- Is there a section on graphing non-simplifiable radical expressions such as the one Sal did around the2:00minute mark?(2 votes)
- At1:37, how does Sal just know that y=square root of (x-3) gives off that curved line. Similarly, with the negative version... It's not even in proper slope-intercept form so how the heck does he know where the hell the line is going. Someone please help!(2 votes)
- At1:37

How do you suppose "y=square root of (x-3)" or the negative version gives you that curved line? That literally makes zero sense to me. He doesn't even have it in slope-intercept form to do that. Someone please explain!(2 votes) - if you squared any input does that automatically give you two potential outputs, because of the potential for positive and negative answers.(1 vote)
- Squaring a number will always give you the same output
`(-3)^2 = 9 = 3^2`

Taking the square root of a number could result in a positive or negative output`√9 = ±3`

(3 votes)

## Video transcript

In the relation x is
equal to y squared plus 3, can y be represented as a
mathematical function of x? So the way they've
written it, x is being represented as a
mathematical function of y. We could even say that
x as a function of y is equal to y squared plus 3. Now, let's see if we
can do it the other way around, if we can represent
y as a function of x. So one way you could
think about it is you could essentially try
to solve for y here. So let's do that. So I have x is equal
to y squared plus 3. Subtract 3 from both
sides, you get x minus 3 is equal to y squared. Now, the next step is going
to be tricky, x minus 3 is equal to y squared. So y could be equal to-- and I'm
just going to swap the sides. y could be equal to-- if we take
the square root of both sides, it could be the positive
square root of x minus 3, or it could be the
negative square root. Or y could be the negative
square root of x minus 3. If you don't believe me,
square both sides of this. You'll get y squared
is equal to x minus 3. Square both sides
of this, you're going to get y squared
is equal to-- well, the negative squared is just
going to be a positive 1. And you're going to get y
squared is equal to x minus 3. So this is a situation
here where for a given x, you could actually
have 2 y-values. Let me show you. Let me attempt to
sketch this graph. So let's say this is our y-axis. I guess I could call
it this relation. This is our x-axis. And this right over here,
y is a positive square root of x minus 3. That's going to look like this. So if this is x is equal to 3,
it's going to look like this. That's y is equal to the
positive square root of x minus 3. And this over here, y is equal
to the negative square root of x minus 3, is going to
look something like this. I should make it a little
bit more symmetric looking, because it's going to
essentially be the mirror image if you flip
over the x-axis. So it's going to look
something like this-- y is equal to the negative
square root of x minus 3. And this right over here,
this relationship cannot be-- this right over here
is not a function of x. In order to be a function
of x, for a given x it has to map to exactly
one value for the function. But here you see it's mapping
to two values of the function. So, for example, let's say
we take x is equal to 4. So x equals 4 could get
us to y is equal to 1. 4 minus 3 is 1. Take the positive square
root, it could be 1. Or you could have x equals 4,
and y is equal to negative 1. So you can't have
this situation. If you were making a table
x and y as a function of x, you can't have x is equal to 4. And at one point it equals 1. And then in another
interpretation of it, when x is equal to 4,
you get to negative 1. You can't have one input
mapping to two outputs and still be a function. So in this case, the relation
cannot-- for this relation, y cannot be represented as a
mathematical function of x.