Definition: An equation is considered as function if for any x in the domain of the equation (the domain is the entire x's which can be plugged into the equation) the equation will yield accurately one value of y.

Usually this is easier to understand with an example.

Example 1 Determine if following are functions.

(a) y = x² + 1

(b) y ²  = x + 1

Solution

 (a) The first one is a function.  Given an x, there is just one way to square it & then add 1 to the result. Thus, no matter what value of x you put in the equation, there is just one possible value of y.

 (b) One difference between this equation & the first is that we moved the exponent off the x & onto the y. This small change is all which is required, in this case, to alter the equation from a function to something which isn't a function.

To see that it isn't a function is fairly simple.  Select a value of x, say x=3 and plug this into the equation.

y ²  =3 + 1 =4

Now, there are two probable values of y which' we could utilize here. We could use

y = 2 or y = -2 .

As there are two probable values of y which we get from a single x this equation isn't a function.

Note that this only has to be the case for a single value of x to build an equation not is a function.  For example we could have utilized x=-1 and in this case we would get a single y (y=0).

Though, Due to what happens at x=3 this equation will not be a function.