# What is a "function"?

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• jenniferAOI

#### jenniferAOI

why does a function have to be called function? Can't it be just be called a relation, for instance, of 2 variables?
the word function is confusing me. Of course, I have searched and read its definition but i just can not get it. I could just set it in my head but I don't understand it well enough.

why does a function have to be called function? Can't it be just be called a relation, for instance, of 2 variables?
the word function is confusing me. Of course, I have searched and read its definition but i just can not get it. I could just set it in my head but I don't understand it well enough.

A function is a special relation:

A function f: A→B: x→ f(x) is a relation where given an x ∈ A, you can only find one f(x) ∈ B. We call x the variable, since f(x) depends on the value of x.

libervurto
A function is a special relation:
thanks for that. then how about the function which is like a "machine thing"? In what sense exactly?

thanks for that. then how about the function which is like a "machine thing"? In what sense exactly?
You mean a composition of functions? Is that what you mean by saying "machine thing"?

thanks for that. then how about the function which is like a "machine thing"? In what sense exactly?
You can think of a function as a sort of machine in which one number goes in (the input), and another comes out (the output). For example if you have a function whose formula is f(x) = 2x - 3, an input value of 5 results in an output value of 7. In function notation, this is f(5) = 2(5) - 3 = 7.

You mean a composition of functions? Is that what you mean by saying "machine thing"?
I don't believe she is talking about function composition here.

jenniferAOI
The "machine thing" is a good way to understand "function". You put an element into the machine, the machine does some process to the element, and something comes out of the machine, some item which is a result which depends on the item input and the stuff inside the function.

jenniferAOI
I agree fully with you

why does a function have to be called function? Can't it be just be called a relation, for instance, of 2 variables?
the word function is confusing me.

It may help to distinguish between "common language" and mathematical terminology. For example, an article on economics might say "job creation is a function of the prime interest rate", which in common language means that the number of jobs created is affected by the prime interest rate. However such a statement wouldn't be interpreted as a claim about a mathematical function - i.e. it wouldn't be a claim that if you are given the exact value of the prime interest rate (e.g. 3.02 %) that you could could determine the exact number of jobs created (e.g. 314,255 ).

jenniferAOI, PeroK, ProfuselyQuarky and 1 other person
A function being something different than a relation is that a function will give only ONE result for any input. A relation can give more than one result for anyone input. The distinction is made to help identify possible reversibility.

jenniferAOI
It may help to distinguish between "common language" and mathematical terminology. For example, an article on economics might say "job creation is a function of the prime interest rate", which in common language means that the number of jobs created is affected by the prime interest rate. However such a statement wouldn't be interpreted as a claim about a mathematical function - i.e. it wouldn't be a claim that if you are given the exact value of the prime interest rate (e.g. 3.02 %) that you could could determine the exact number of jobs created (e.g. 314,255 ).
Well said, I really like how you wrote this :)

A function is a relation between two sets ##A## and ##B##, which might contain any type of object you like, but with the restriction that each object of ##A## is in relation with at most one object of ##B##, which is zero or one object. The definition set of a function is the subset of ##A## that contains all elements of ##A## that are in relation with exactly one element of ##B##.

jenniferAOI
For example if you have a function whose formula is f(x) = 2x - 3, an input value of 5 results in an output value of 7. In function notation, this is f(5) = 2(5) - 3 = 7.
I understand. Is the one we are calling the "function" is the formula itself?

It may help to distinguish between "common language" and mathematical terminology. For example, an article on economics might say "job creation is a function of the prime interest rate", which in common language means that the number of jobs created is affected by the prime interest rate. However such a statement wouldn't be interpreted as a claim about a mathematical function - i.e. it wouldn't be a claim that if you are given the exact value of the prime interest rate (e.g. 3.02 %) that you could could determine the exact number of jobs created (e.g. 314,255 ).
yes and this is where the
A function is a special relation
comes in.The word "function" relates two things in a special way which applies in both languages. Just that in mathematics, it demands an exact value while in your economics example it was a mere statement of the relationship between the 2 things, right?

A function being something different than a relation is that a function will give only ONE result for any input. A relation can give more than one result for anyone input.
thank you. This is a great help.
The distinction is made to help identify possible reversibility.
but what kind of reversibility?

thank you guys for all the responses. I really appreciate it. It is somehow becoming clearer. But another question:
In calculus, why does it uses just function and not relation?

thank you. This is a great help.

but what kind of reversibility?
thank you guys for all the responses. I really appreciate it. It is somehow becoming clearer. But another question:
In calculus, why does it uses just function and not relation?
We want to be able to UNDO what a function does.
f(x) can be a function. You put in a number x, and the function gives output y.

We also want some function maybe called g(x), so that g(f(x))=x and f(g(x))=x.

We want to be able to UNDO what a function does.
f(x) can be a function. You put in a number x, and the function gives output y.

We also want some function maybe called g(x), so that g(f(x))=x and f(g(x))=x.
okay. So there ain't any reversing the process in a relation?

okay. So there ain't any reversing the process in a relation?
A function is still a relation; but not every relation is a function.

The better way to learn this slow and deep is to study Intermediate Algebra. That will make some ideas clearer.

jenniferAOI
A function is still a relation; but not every relation is a function.

The better way to learn this slow and deep is to study Intermediate Algebra. That will make some ideas clearer.
Okay. Thank you. :)