How can you determine if a relationship is a function?

Question

How can you determine if a relationship is a function?
Explain please

Answer 1

If I might just add one more example to the excellent previous posts:

x^2 +y^2 = 25 is not a function, because if, for example, you choose x=3, there are TWO values of y that work: +4 and -4.

If you solve for y in terms of x, you get y = "plus or minus" the square root of 25-x^2. Two values of y.

Answer 2

A relationship is a function if there are no x values that lead to two different y values.

For example:
(1,2) (1,3) (2,7) (3,9) is not a function because 1 leads to both 2 and 3.

It's OK if different x values lead to more than one y value.
(1,2) (2,2) (3,2) (4,2) is still considered a function because there is no x value that leads to different y values.

An easy way to tell if a graph is a function is by using the vertical line test. If you place a vertical line anywhere on the graph and it passes through more than 1 point, it is not a function.

Answer 3

A relation is a function if for all x, y,z,
if (x,y) and (x,z) are elements of that relation
then y=z.

That's the definition of a function.