Question

Which of these equations represent functions?
A. y= x^2
B. y^2= x^2
C. x^2-y^2=1
D. y= 3x

Answer 1

Ah, well let us begin by defining what is a function and what is not a function, shall we?

Answer 2

A function is a mapping from a set of values to another set of values so that every input has exactly one output.

Two inputs may be mapped to the same output, but in no case is it allowed that 1 input be mapped to two separate outputs.

Answer 3

Following me?

Answer 4

I am now

Answer 5

Okay good.

Answer 6

So let's look at the functions given. The first one is y = x^2
Does it meet the requirements of a function? Is there any x that gets mapped to 2 different ys?

Answer 7

I don't think there is

Answer 8

If the x is mapped to two different ys, it's not a function, right?
Or is it the other way around?

Answer 9

Correct. If even one x gets mapped to two different ys, then it's not a function.

Answer 10

And like you said, for that first part, there are no xs that get mapped to two ys.

Answer 11

So would A be one of the correct answers?

Answer 12

Yes.

Answer 13

So now let's look at B. y^2 = x^2.

Answer 14

How can I determine by looking at the equation whether or not one x maps to two ys?

Answer 15

Well... just start by looking at an x value or two. Pick x = 2 for example.

Answer 16

x^2 = 4, so are there multiple ys or just 1 y that satisfy the equation y^2 = 4?

Answer 17

There's only one y in the equation for B, so that works as a function, right?

Answer 18

I'm thinking that all except D are correct

Answer 19

Let's slow down and look at B.

Answer 20

So for x = 2, y = 2 satisfies the equation y^2 = x^2, right?

Answer 21

I'm thinking yes

Answer 22

Yes. It does. So 2 gets mapped to 2.

Answer 23

But is there another y so that y^2 = 2^2?

Answer 24

I don't see another y

Answer 25

So it only gets mapped to one y

Answer 26

No? Tell me. What if y = -2? What is y^2 then?

Answer 27

-4?

Answer 28

I'm not very good at Algebra 2, lol

Answer 29

(-2)*(-2) = -4? You're multiplying two negative numbers.

Answer 30

You're doing great. Just stick with me =) Don't get discouraged.

Answer 31

So it would be a positive 4 instead?

Answer 32

Yes. So for y= -2, y^2 = 4 = x^2, so y^2 = x^2, right?

Answer 33

Which means what?

Answer 34

There's more than one y, so it wouldn't be a function?

Answer 35

Yes. So we looked at x = 2, saw that x^2 =4 and wondered what y values would work then. And we saw that for both y = 2 and y = -2, y^2 = x^2 worked. Which means that x = 2 gets mapped to both 2 and -2.

Answer 36

Get it?

Answer 37

I think so. So B isn't an answer

Answer 38

Right. Let's look at C.

Answer 39

x^2 - y^2 = 1

Answer 40

Look at it for a bit, maybe at one or two xs and tell me what you think.

Answer 41

I'm not coming up with anything by looking at it, lol.

Answer 42

Wait, there would be 2 y's since the y is squared?

Answer 43

Yes. Let's see if we can come up with an example.

Answer 44

What about when x is 2?
x^2 - y^2 = 1 so
4 - y^2 = 1 so
-y^2 = -3 so
y^2 = 3
What values for y work in this case?

Answer 45

sqrt(3) works. And?