1. When I see the word mathematics, I just remember the function which I learned in high school.
Actually, It is a little hard for me. Because, in primary school and junior high school, My mathematics is not good.
Are you guys interested in Function?

2.

3. Are you referring to functions, such as or are you referring to calculus in general?

4. Well, you can think of a function as a kind of machine that takes in a certain object and outputs another object according to some rule encoded in the machine.

It is considered good practice to specify what type of object is being input and what type is being output. So for example one writes , where X and Y are defined to contain certain specified types of object.

In the above, is called the domain and is called to codomain (sometimes called the range space). So the object is said to be the image of the object .

Let's now insist on 2 rules for well-behave functions

1. No function can have more than one image in its codomain

2. Every function must have an image in itscodomain. This can give rise to some difficulty unless we are careful. Consider the function . This is clearly nonsense when , so we need to identify the domain as not containing the zero element. There are several notational conventions for this

There is another notational convention that I will get to after this - in what follows please take my words completely literally:

A function is said to be injective if, fo all there is at most one such that

A function is said to be surjective if, for all there is at least one such that

A function is said to be bijective if, fo all there is at least one and at most one such that .

So, those objects for which is called the pre-image of , written . Note from the definition of a surjective function this may be a multiplicity of objects, so the pre-image is a subset of , and therefore this notation need not imply the "inverse function".

This is also true of bijective functions, even though the pre-image is a single object. Strictly we should call it the "singletion set" but nobody ever does - just write [tex]x/tex] and so and are taken to be mutual inverses.

Anything else? Yeah, loads, but that is the bare bones

5. Yeah, I refer to the former one.
Originally Posted by Cogito Ergo Sum
Are you referring to functions, such as or are you referring to calculus in general?

6. It's very specific, but I cannot understand it thoroughly....
Originally Posted by Guitarist
Well, you can think of a function as a kind of machine that takes in a certain object and outputs another object according to some rule encoded in the machine.

It is considered good practice to specify what type of object is being input and what type is being output. So for example one writes , where X and Y are defined to contain certain specified types of object.

In the above, is called the domain and is called to codomain (sometimes called the range space). So the object is said to be the image of the object .

Let's now insist on 2 rules for well-behave functions

1. No function can have more than one image in its codomain

2. Every function must have an image in itscodomain. This can give rise to some difficulty unless we are careful. Consider the function . This is clearly nonsense when , so we need to identify the domain as not containing the zero element. There are several notational conventions for this

There is another notational convention that I will get to after this - in what follows please take my words completely literally:

A function is said to be injective if, fo all there is at most one such that

A function is said to be surjective if, for all there is at least one such that

A function is said to be bijective if, fo all there is at least one and at most one such that .

So, those objects for which is called the pre-image of , written . Note from the definition of a surjective function this may be a multiplicity of objects, so the pre-image is a subset of , and therefore this notation need not imply the "inverse function".

This is also true of bijective functions, even though the pre-image is a single object. Strictly we should call it the "singletion set" but nobody ever does - just write [tex]x/tex] and so and are taken to be mutual inverses.

Anything else? Yeah, loads, but that is the bare bones

7. Originally Posted by Guitarist
Well, you can think of a function as a kind of machine that takes in a certain object and outputs another object according to some rule encoded in the machine.
There are way more functions than there are rules. A better metaphor is that a function is a machine that transforms an object in a black box. We can know NOTHING about the mechanism. All we know is that whenever we put in a particular input value we always get the same output value. But that's all we know. The mechanism is unknown.

I think that's a better way to explain this to beginners. Because it's true to the spirit of what mathematical functions are; and because it makes you think a little. What kind of functions do have rules? What kind of functions don't have rules?

Anyway I'm sure your post was very helpful to the OP :-)

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