Why Do Functions Have Only One Output for Each Input?

In summary: This distinction helps us understand the properties and usefulness of functions in a clearer way.In summary, the distinction between functions and relations lies in the fact that functions have only one possible y value for each x value, while relations can have multiple possible y values for the same x value. This distinction is important as it helps us understand the properties and usefulness of functions in a clearer way.
  • #1
Rishabh Narula
61
5
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?
 
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  • #2
Rishabh Narula said:
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.
 
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  • #3
Rishabh Narula said:
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.
 
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  • #4
Rishabh Narula said:
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.
 

Related to Why Do Functions Have Only One Output for Each Input?

What is a function?

A function is a mathematical concept that represents a relationship between two variables, where each input has exactly one output. It can also be thought of as a rule that converts one value into another.

What is the difference between a function and a relation?

A function is a type of relation where each input has exactly one output. In other words, each input value is mapped to only one output value. A relation, on the other hand, can have multiple outputs for a single input.

What is the domain of a function?

The domain of a function is the set of all possible input values for which the function is defined. It is the independent variable in a function and is usually represented by the letter "x".

What is the range of a function?

The range of a function is the set of all possible output values that the function can produce. It is the dependent variable in a function and is usually represented by the letter "y".

What is the difference between a linear and a non-linear function?

A linear function is a function whose graph is a straight line, meaning that the rate of change between input and output values is constant. A non-linear function, on the other hand, is a function whose graph is not a straight line, meaning that the rate of change between input and output values is not constant.

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