Why Do Functions Have Only One Output for Each Input?

[Rishabh Narula]

Why do we define functions as only
as only those graphs which have
only one y value for each x value.
for eg. we don't say that a circle
is a graph of a function,because
its graph would have two y values
for same x values.
what i mean to ask is why not call
anything that takes a input and gives
an output a function?Why this distinction?

 

[PeroK]

Why do we define functions as only
as only those graphs which have
only one y value for each x value.
for eg. we don't say that a circle
is a graph of a function,because
its graph would have two y values
for same x values.
what i mean to ask is why not call
anything that takes a input and gives
an output a function?Why this distinction?

The question is really why are functions so useful? There are a lot of examples where what we want to study is something that has a definite value. The position of an object at time $t$ is a function of $t$.

The function represents this relationship of one input, one output, which is very useful.

 

[fresh_42]

what i mean to ask is why not call
anything that takes a input and gives
an output a function?Why this distinction?

Because we have already have a name for it. They are called a relation. Functions are special relations with only one possible y value. It's easy to see that this is more useful than a relation. If you compute the square root of a number, then your algorithm will only give you one solution, not both. If we drive from A to B, we will be at only one place at time T, not two; etc. And as I said: in case we are interested in a one to many relation, we will call it a relation. We have a name.

 

[FactChecker]

why not call
anything that takes a input and gives
an output a function?

I guess you mean "a choice of outputs" because that is what your example of the circle does.

Why this distinction?

Because it is important to distinguish between the case where it gives a single output from the case where it gives a choice of multiple outputs.