What equation is not a function?

Answer 1

There are a few ways to know a relation between x and y is not a function, the hardest of which is looking at the equation without knowing what the graph looks like. But I always tell my students a couple hints to look for: If the y has an even exponent or is enclosed in an absolute value, it's not going to be a function. This is because with one input for x, you will get more than one output for y, thereby defying the definition of a function. If you know what the graph looks like, it's simple, it has o pass the vertical line test, which means no vertical line can intersect in more than one point.

Answer 2

The easiest example is the equation of a circle x^2+y^2= r^2.
If you do the vertical line test and move a vertical line along the x axis from left to right, you will see that the line cuts the circle at two places for each value of x, giving two values of y for each value of x.. A function can only have one value of y for each value of x.