lugreget9

Answered

2021-12-21

I have been quite confused by the definition of functions and their uses. First of all can one define functions in a clear understandable way, with a clear explanation of their uses, how they work and what they do?

Also I have some specific questions regarding functionsLet me give you a examples:

Also I have some specific questions regarding functions

Let me give you a examples:

1.$y=f\left(x\right)\Rightarrow$ This is one of the main reasons I have difficulties understanding functions...

What does the above statement tell me, and if y is a function why do we use$y=$ at all for a formula like $y=mx+b$ would it be the same as writing $f\left(x\right)=mx+b?$

2. Something like$y={x}^{2}$ is apparently a function... but where is the function name? Which is the input and which is the output?

3. Lastly another example: Let me suppose$f=$ distance $f\left(t\right)={t}^{2}$

$f\left(2\right)=4\Rightarrow$ Does this mean distance is 4... which is the input which is the output?

Mainly what I'm looking for in an answer is a clear and 'easy' explanation with examples to help me understand this new topic

Also I have some specific questions regarding functionsLet me give you a examples:

Also I have some specific questions regarding functions

Let me give you a examples:

1.

What does the above statement tell me, and if y is a function why do we use

2. Something like

3. Lastly another example: Let me suppose

Mainly what I'm looking for in an answer is a clear and 'easy' explanation with examples to help me understand this new topic

Answer & Explanation

Dabanka4v

Expert

2021-12-22Added 36 answers

I see there are already a few answers to this question but I'm going to try my hand at answering it the way I see it.

A function f is a mathematical object that relates elements of two sets, one called the domain A and one called the codomain B. The notation

What it means to be a function

Those elements of B which can be written in the form f(a), for some

here are various ways of specifying functions. For example:

If A is finite, you can simply list the values of f. For example we can define a function

Sometimes functions can be defined by an equation. An example of a function

This equation is not itself a function. What it means is, given an element

The graph of a function

Sometimes, particularly when

That is, the function f specified by this equation is the one which associates to each

An example of how a function works is as follows. Suppose a bird is flying in a straight line at a constant speed of 12 metres per second. The distance the bird flies 'is a function of time', in the following sense: if t is a positive real number, then the distance flown by the bird in t seconds is 12t metres. Thus the relationship between distance and time defines a function

for all

Some non-examples of functions are:

In summary, if

f is the function itself, which has domain A and codomain B;

In the mathematical branch of set theory, which is used as a foundation for most mainstream mathematics, we need to specify precisely in terms of sets what it means for f to be a function. In this setting, a function

This formal approach isn't something you need to worry about if you're learning about functions for the first time. All that matters is that for every element of the domain A, f identifies that element with exactly one element of the codomain B.

This explanation is woefully incomplete, but there's only so much you can do in an MSE answer... let me know if you need more clarifications.

trisanualb6

Expert

2021-12-23Added 32 answers

A function requires some inputs and for each valid combination of inputs produces one output. What is valid is determined by the domain, which is sometimes specified but sometimes left for the reader to infer. The issue is when talking about graphs, because historically people have used single letters to refer to changing quantities, and still do so in many areas of mathematics. When we say that "y is a function of x" it means that "y changes together with x". That concept does not correspond directly to the proper definition of functions.

As you realize, when you have a straight line it can be described by the linear function f where

To use your other example, if f is a function such that

Note also that the variable used as the input in defining a function is not important and is called a dummy variable. In the above example we could have defined f to be such that

So "

Moreover, some graphs (and their associated equations) do not even correspond to a function, such as "

For a concrete example, if say you are told that the position of a ball above the earth's surface is

For an example of a function with multiple inputs, consider a function d such that

For another example, consider max such that

In general a lot of things in mathematics can be thought of as functions.

Finally, I did not talk about set theory because that is not the only way of defining a function and is in my opinion not the most intuitive way. What I said above holds whether functions are defined in terms of sets or not.

nick1337

Expert

2021-12-28Added 573 answers

I'm going to take a little bit of a different approach from that commonly found in textbooks, and instead of giving you mathematical explanations and examples, I'll try to explain it in terms of the semantics and usage of functions. It seems to me that this where the confusion actually lies (not just for you - for most other people too.)

First off, the definitions others have posted here and that you probably can find in your textbook are correct. I'll put it into layman's terms:

That's a pretty broad definition, which probably doesn't sound too useful, and which also probably doesn't go far in explaining all of the

So let's think about this definition a little bit first. What we are talking about is an abstract idea - we are talking about the concept of a "thing" which takes in information and gives back some other information. What makes a function special is the requirement that it has to be predictable - one input can't give two or more different outputs.

This isn't too hard of an idea to grasp - you are already familiar with lots of things that do that. For example, 2 + 2 = ? You know the answer to this is four - you look at the + symbol, then look at the inputs: 2 and 2, and you know you are supposed to add them together. Here the + symbol is an operator that represents the addition function. This is a language used to represent the abstract concept of addition, and we use it so that we can communicate the idea of adding two numbers together to get a sum in a concise manner.

I think this is where you are getting stuck. You know that if you have something like:

y=mx+b

that you can "plug in numbers" for m, x, and b, then follow the order of operations to get an answer for y = ?

You also said that we can just write f(x)=mx+b, which is true. Why don't we? The answer is: it's just semantics. The concept of taking some numbers, plugging them into an expression, and evaluating to get a single answer is the function. y=mx+b is a representation of the function - it's a convenient, concise notation. The same is true for f(x)=mx+b

So why do we have both notations, and what is their use? Well, the one you are used to: y=mx+b is good when y represents some quantity, measurement, or other thing that you understand pretty well. For example, this is a linear equation, and since we typically plot lines on Cartesian axes and label them x and y, in this case y would represent the y-coordinate of points on the line. We usually graph independent variables on the x axis, so x is typically our input. m is the slope and b is the y-intercept, and these don't change for a given linear relationship, so we call them parameters - they define the linear function. Since we are used to seeing it written like this in this context, there is not much confusion about what the inputs and outputs are - x is usually the input, y is usually the output. The notation matches up with the common usage, and it's convenient and simple, so we use it.

The new notation - the one that is confusing, with the f(x) [and later the g(x) and h(x) and really anything(anything_else)] serves a slightly different purpose. The f(x) part tells you what the inputs are, explicitly. If I say f(x)=mx+b, I am telling you directly that x is an input. Once I "plug in" a value for x, then whatever the right hand side ends up evaluating to is the output.

So why would we want to use this? For a linear equation, we probably wouldn't, at least not in most cases. When learning this stuff though, we often start with that so that you can see how it all works. There are two major advantages to f(x) notation, though:

1. You can exactly communicate what the intended input variables are

2. You can write an expression for a function even when you don't know what the exact relationship is

2 is very important. This is why you need to learn this notation at this level - because soon you are going to start working with functions in an even more abstract sense - you will do things like this:

"I have this big equation with y in it that I can't solve yet. I know that y is a function of x and t, but I don't yet know what that relationship is. So, I'll let y=f(x,t), then use that as a placeholder while I do a bunch of things like take derivatives and integrate in order to solve the main problem."

In other words, it lets us write down what we do know when we don't know everything, which happens pretty often.

Now you probably have the answers to your three questions from reading that, but just in case it wasn't clear:

It would be the same - we use y when we don't need to be very clear about which variable(s) are inputs.

This function has no name yet - although you could just say "the function y equals x squared." More likely you would say "y is a function of x." The function happens to be

Yes, if distance d is a function of t, and

Most Popular Questions