# How do you decide whether the relation # x^2 + y = 81# defines a function?

##### 1 Answer

#### Explanation:

A function is a law of association between elements in two different sets.

So, to define a function, you must have a "starting" set **domain**), a "landing" set **codomain**), and a rule that associates to **every** element of **one and only one** element of

In most cases, students deal with numerical functions, i.e. functions in which both domain and codomain are the set of real numbers

So, does your equation describes a function? It depends on the variable we choose and independent:

**Case 1: #x# is independent**

In this case, we're looking for a rule that assigns a value to

which is indeed a function: for every value

**Case 1: #y# is independent**

This case is the other way around: we're looking for a rule that assigns a value to

which is **not** a function: for every value

Assume, for example,

So, in this case, we're not able to associate **one and only one** output value to our input value.