Recognizing Functions - Example 4

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.