Question

How do you decide whether the relation `x = y²` defines a function?

Answer 1

We can graph the relation and do the vertical line test.

Explanation:

Whenever we have to check if a relation is a function, we draw its graph and do the vertical line test.

The vertical line test says that if a vertical line touches the graph once, the relation is a function. And if the vertical line touches the graph more than once it's not a function.

Example `->`

`x=y^2`

`sqrtx=y`

`y=sqrtx`

Let's graph this relation.

graph{sqrtx [-10, 10, -5, 5]}

Now if we draw a vertical line we can see that the line touches the graph at only one point everywhere.

Therefore it is a function.

Additional `color(red)(->)`Circles, Ellipses, Hyperbolas etc. are not functions because they don't pass the vertical line test.
A Line, Parabola etc. are functions because they pass the vertical line test.