# Is it possible for two quantities to be related and yet for neither to be a function? How? An example would be very much appreciated!

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Your question isn't well constructed. Quantities cannot be functions -- quantities are objects (numbers) while functions are operations on sets of objects.

If you meant to ask if it is possible for two sets to have a relation that is not a function, then the answer is yes.

Suppose you have two sets, X and Y, both containing all of the real numbers from -4 to 4 inclusive. Then the relation `x^2+y^2=16` where `x in X,y in Y` (x comes from the set X, y comes from the set Y) is certainly not a function. Note that when x=0 y can be either 4 or -4. This cannot happen in a function (each input is paired to only one output). ((This relation is a circle centered at the origin with radius 4: besides the algebraic argument that it is not a funtion, the graph fails the vertical line test))

You can create two finite sets and a relation that is not a function: Let `X={1,2},Y={1,2,3}` and the relation be `{(1,1),(1,2),(2,2),(2,3)}` . This relation is not a function: 1 is paired to 1 and 2 while two is paired to 2 and 3.