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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 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.