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# Checking if an equation represents a function

## 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 two squared is just going to be a positive one and you're going to get Y squared is equal to X minus three so this is a situation here where for a given X you could actually have two Y values let me show you so let's say let me attempt to sketch this graph so let's say this is our y-axis let's get 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 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 it's going to look something like this it's going to look something it should make 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 is equal to the negative square root of x minus three and this right over here this relationship cannot be this right over here is not a function of X for given 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 to make 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 at another interpretation of it when X is equal to 4 you get to negative 1 you can't have one input mapping to 2 outputs and still be a function so in this case the relation cannot what for this relation Y cannot be represented as a mathematical function of X