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

1 Answer
May 29, 2018

#y = f(x) = -x^2+81# is a function

#x = f(y) = \pm\sqrt(-y+81)# is not.

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 #A# (the domain), a "landing" set #B# (the codomain), and a rule that associates to every element of #A# one and only one element of #B#.

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

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 #y#, given a value for #x# as input. Subtracting #x^2# from both sides of the equation, we get

#y = -x^2+81#

which is indeed a function: for every value #x# we may choose, the correponding #y# will be #-x^2+81#

Case 1: #y# is independent

This case is the other way around: we're looking for a rule that assigns a value to #x#, given a value for #y# as input. Subtracting #y# from both sides of the equation, we get

#x^2 = -y+81#

which is not a function: for every value #y# we may choose, assuming #-y+81# is positive, we have ambiguity on the correponding #x#.

Assume, for example, #y=0#. The equation would be #x^2 = 81#, which yields #x=\pm9#.

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