MATH: Why use a function instead of another variable?

[Gogf]

So we've just started learning calculus, and one thing that we've done a lot lately is doing operations to functions. h(x) = f(x)*g(x), for example. Of course, one shorthand we use is h = fg, which makes sense, but it's got me wondering: why include the (x) in the first place? Why can't f(x) = x^2 just be f = x^2?

 

[mangxema]

You could... but it works out better that way when you're considering multiple dimensions. ie, accelerations: a(x), a(y), a(z). Same variable name, more info. You could use subscripts and stuff but that gets really messy really fast.

 

[Aramazd]

You wouldn't know what variable your using if you used f instead of f(x).

 

[mangxema]

I believe he's thinking of the 1-D case where the ind. variable is understood to be something in particular.

 

[Gogf]

I'm thinking of 2D cases, such as a classic Cartesian coordinate plane. Makes sense when you consider multiple variables though... thanks.

 

[Perfection]

A function has the defining property of giving a single specific output to any input in the function's domain something simple relations lack. It is true that sometimes this property is unimportant for certain applications, but there are many times when it is not.

 

[Atticus]

You can sometimes left the (x) out, but it requires some judgment, so it's better not to until you're confident with that.

The thing is that x marks the variable, you read "f as a function of x". It makes very much sense, if you have constants in the function.

Example 1: You are working t hours with a salary of m dollars per an hour. Your earnings as a function of time worked is
e(t)=tm.
As a function of your hourly salary it is
e(m)=tm.
In the function above you think m as a constant and t as a variable. In the function below t is a constant and m is variable.

Example 2: If you drop ball from the altitude of a meters at the moment t=0, it's altitude will be f(t)=a-1/2 gt^2 (here g is the gravitational constant). Now the (t) part says, that you are talking about ball's altitude as a function of time.

Example 3: You change the original altitude of ball, and are interested about it's altitude one second after the drop, now the variable is a:
g(a)=a-1/2 gt^2 =a-1/2 g

(Yes, the ball will hit the ground at some moment, and that means these functions aren't applicable for all t:s and a:s, but let's not go into that).

Note that as a courtesy to all you yankees out there I used dollars as the unit of money!!!!

 

[nihilistic]

Why can't f(x) = x^2 just be f = x^2?

Well, those two statements say different things. The first defines only f(), as a function that maps any x to x^2. The x in the first function is essentially invisible. The second expression defines f and x as variables with the relation between them. Very different.

The first expression tend to be more flexible in many many cases, because the same function can be applied to other variable. With the second method, if you want to describe a similar relation regarding some other variable, say, y, you'd have to write down something like g = y^2. With the first notation, you could describe that function simply as f(y), as you have already described the function. Then there are other possibilities, such as f(f(x) - 3), which of course would be understood as (x^2 - 3)^2.

It also makes it clear that you are not studying the behavior of the variable (which is usually very boring, probably the real line or something, unless you are taking real analysis or beyond), but the behavior of the function.

Also, functions can have more than one variable. From then on it is absolutely essential to know exactly what the function has as inputs.

 

[Xyan]

2 Reason:

1. There could be more than one variable for more complex functions. Or even coordinates of a point which most students should be familiar with!

2. You will want to know which variable you are dealing with. Other posters have mentioned this. I would add that this becomes even more important when you start handling inverse functions.

 

[History_Buff]

It also becomes more important when you're doing vector calculus, and you need to know exactly what is dependant on what other variables, so you can do your chain rule right.

But essentially it all comes down to defining input variables. For instance, if you had

h(x,y) = f(x)*g(y)

Then dh/dx is simply f'(x). Something that may not have been obvious if you had merely written h = fg

 

[linfeixb27]

It also becomes more important when you're doing vector calculus, and you need to know exactly what is dependant on what other variables, so you can do your chain rule right.

But essentially it all comes down to defining input variables. For instance, if you had

h(x,y) = f(x)*g(y)

Then dh/dx is simply f'(x). Something that may not have been obvious if you had merely written h = fg

I think it should be