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

Question

Isaiah Anthony · Accepted Answer

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

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

A law of association connecting elements in two distinct sets is called a function. 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}#. Thus, depending on the variable we select and the independent variable, does your equation describe a function? 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#. Therefore, we are unable to correlate our input value with a single output value in this instance.

Eva Adamson · Answer

undefined To determine if the relation $x^2 + y = 81$ defines a function, we need to check if each input value (x) corresponds to exactly one output value (y). We can rewrite the equation as $y = 81 - x^2$. Since the output value (y) is dependent on the input value (x), and for each value of x, there is only one corresponding value of y, the relation defines a function.