f(x) = y

[cscott]

Can someone please explain what f(x) = y means? I know it's function notation, but what does it imply?

[Callipygous]

they are the same thing. thats a definition, not any equation. im not sure what you mean by what does it imply.

[cscott]

I just don't understand how it can make sense to use it in the general rule for differentiation. How can it represent a function?

 

I don't know if I make sense...

[CPL.Luke]

if I understand your question correctly you are asking why you can substitute f(x) for y right?

 

the reason for this is that you already said that the y variable is equal to some function of x. or in other words you defined a relationship between the x and y variables

[ecoli]

y and f(x) are essentially interchangable.

 

[math]y = x^2 [/math] and [math]f(x) = x^2[/math] is the same thing..

 

For example if you used the above equation and you wanted to find out what would the y value be if X = 2, you could express the equation like this:

[math] $If $ f(x) = x^2$, $ f(2) = ? [/math]

[cscott]

if I understand your question correctly you are asking why you can substitute f(x) for y right?

 

the reason for this is that you already said that the y variable is equal to some function of x. or in other words you defined a relationship between the x and y variables

 

If the statement means f(x) is a substitute for y' date=' then why is it ([math']y = f(x)[/math]) used as a function when finding [math]\lim{x \rightarrow 0} \frac{f(x + \Deltax) - f(x)}{\Delta x}[/math] where you add [math]\Delta y[/math] to y and [math]\Delta x[/math] to f(x)? This is for the general rule for differentiation again...

 

What am I missing?

[ecoli]

well, that isn't the general rule for differentiation, you missing "h"'s.

[matt grime]

Can someone please explain what f(x) = y means? I know it's function notation, but what does it imply?

 

 

this is a good example of a simple questiopn that is quite hard to answer.

 

right, let X and Y be two sets. a function is a way of assigning, unambiguously, an element of Y to each element of X. Unambiguous means that x in X gets assigned to exactly 1 one element of Y.

 

they way we write this formally is to store the information as ordered pairs. if you prefer you can think of these as coordinates (u,v) where u is in X and v is the element of Y assigned to it. Sometimes we choose to describe v as being a funtion of u by writing v=f(u) where f is another way of writing the function. so f is telling you how the elements of X and Y are related.

 

it implies nothing, it is just notation.

[zking786]

Here's a long shot:

y is a function of x (for every x value, there is only one y-value). Because an equation y=.... doesn't have to be a function, it can have multiple y-values for a single x-value. Clarifying y=f(x) makes us certain that y is a function. Is this what you mean?

[zking786]

Oops! I missed your question...

you want to take the limit of (F(x+deltaX)-F(x))/deltaX as deltaX approaches 0

[cscott]

this is a good example of a simple questiopn that is quite hard to answer.

 

right' date=' let X and Y be two sets. a function is a way of assigning, unambiguously, an element of Y to each element of X. Unambiguous means that x in X gets assigned to exactly 1 one element of Y.

 

they way we write this formally is to store the information as ordered pairs. if you prefer you can think of these as coordinates (u,v) where u is in X and v is the element of Y assigned to it. Sometimes we choose to describe v as being a funtion of u by writing v=f(u) where f is another way of writing the function. so f is telling you how the elements of X and Y are related.

 

it implies nothing, it is just notation.[/quote']

 

I think I understand now, but I need someone to tell me if the following is true: y = f(x) denotes that y is a function of x but it does not show how they relate (ie. it's not the same as y = x, it could be anything).

[Thomas Kirby]

Disclaimer: I am not an expert.

 

I think that this is what cscott is looking for:

 

[math]y=x*3[/math]

[math]f(x)=x*3[/math]

[math]y=f(x)[/math]

 

 

 

[math]y=f(x)[/math] does not show how they relate but you can go back to the definition of [math]f(x)[/math]. [math]f(x)[/math] is a token, or a label for the equation that you have labelled as f(x). You can use anything in place of the "f" as long as it doesn't cause confusion when you work out your equations. This lets you do things like this:

 

[math]f(x)=x*3 [/math]

 

[math]g(x)=x^2+2[/math]

 

[math]y=(f(x)*g(x))^2[/math]

 

 

It's an easy way to combine whole equations, treating them as variables. That way you don't have to multiply them out in expanded form until you are ready. You might also choose instead to multiply the results of the equations instead of multiplying the equations together.

[matt grime]

I think I understand now, but I need someone to tell me if the following is true: y = f(x) denotes that y is a function of x but it does not show how they relate (ie. it's not the same as y = x, it could be anything).

 

f could be anything, or f could be given as some specific function. it would depend on the question, just like x can be 2 if we specify x=2 or x can be a real number.

 

so y=f(x) is the same as y=x when f is the identity function.

 

remember almost no functions can be written nicely, eg sin(x) or soemthing, so we often specify at the start what f(x) is, and then use f to refer to it for the rest of the discussion.

 

eg, suppose f is defiend such that f(x) is 1 if x is rational and 0 otherwise then... some discussion of f

 

we woulldn't like to have to specify what f is in this case every time would we?

[cscott]

Thanks, you all cleared that up very well... it makes so much more sense now.