Question

How do you decide whether the relation `y^2 =4x` defines a function?

Answer 1

See explanation below

Explanation:

A function is an application such that, for EACH `x` the image is UNIQUE.

In our expresion `y^2=4x` if we apply square roots in both sides we have

`y=+-sqrt(4x)` in this expresion, for each `x` we have two values for `y` (negative and positive), So the expresion doesn`t define a function

A similar approach working with graphs. If we draw a vertical line, (parallel to y axis) this line cut or intercept graph in only one point in plane. If vertical line intercept in two or more points, the graph doesn't define a function.

Hope this helps

Note: the relation above define a parabola (plane curve, not function) like this
graph{y^2=4x [-10, 10, -5, 5]}