determine whether or not the equation represents y as a function of x

Question

determine whether or not the equation represents y as a function of x

Q: X^2=Y^2

I have to show the work

Answer 1

To be a function, it has to pass the "vertical line test" which means for each x-value there is at most one y-value.

 

So,

 

let's rewrite it as y² = x².

Now if we want it to be a function of y we have to reduce it to just the first power of y. Here we can do that by taking the square root. So, you get

sqrt(y²) = sqrt(x²).

 

Now if you want to take square roots of anything that is a square (which means it's of the form z²) you must consider both the positive and negative values, since z² = (-z)² , and in this case y² = (-y)².

 

so then when we reduce the equality sqrt(y²) = sqrt(x²) we get

y = +/-sqrt(x²). When we reduce sqrt(x²) it's a bit redundant to put +/- y = +/- x. So we leave it as

y = +/- x.

 

Now that it's an equality with just y isolated, we test whether it's a function.

so let's take x = something arbitrary, let's say 1.

then y = 1 and y = -1. Clearly that does not pass the function test (or vertical line test as I mentioned earlier).