F(x), G(x) Etc

[roger]

Dear friends,


please can you help me to understand what f(x) means?

I am just starting out at high school maths level.....

In a book it also says g(x) h(x) etc what does it mean ??

Also does the x in the brackets ever change and if so what does it mean ?


Thankyou for any advice.


Roger

[dav2008]

f(x) and g(x) (or any other letter) are notations for functions.

f(x) is read as "eff of x".

let's say that f(x) = x+3.

that means there is a function of x that takes the input and adds 3 to it, so f(3) =3+3.

the f or g or h are just standard letters for representing functions.

You can have
f(x) = x2
g(x) = x-2

So if you want to find f(3) it would be 32=9.
g(3) would be 3-2=1

Edit: Typo, put a 3 instead of 2

[brewnog]

^ what he said.

And also, for all intents and purposes:

f(x) = 3x

means *basically* the same thing as:

y = 3x

as you may already be familiar with this notation.

[Hurkyl]

And also, for all intents and purposes:

f(x) = 3x

means *basically* the same thing as:

y = 3x

Ugh, don't say that. :cry: It might help him for a little while, but can only lead to confusion later.

[cepheid]

Why? Do you mean because it's only true if y = f(x)? i.e. that y and f(x) could be totally different things? Or is there some other more fundamental difference between a dependent variable (whose dependence can be described as a functional relationship) and a function itself that I should have picked up on a long time ago?

[brewnog]

I'm sorry. With all due respect to roger, the questions he's asking are at a pretty fundamental level (relative to a lot of the stuff on this forum in any case), and I thought that my simple (if not entirely accurate) analogy might serve him well, if only for the time being. I remember clearly being taught this new notation years back, and I definitely thought of it this way until I got my head round it all. It's also the reason I put "basically" in nice sparkly asterisks :)

Sorry for any confusion roger!

[Hurkyl]

Sometimes using "shortcuts" does help, but you always have to remember their dangers: it may stifle one's ability to use the thing without the shortcut, or worse believe the thing is the shortcut.


I guess the reason for my reaction is that the biggest problem I've seen happen is understanding evaluation: they could tell you what f(x) is, or f(4), maybe what f(t) is, but be entirely stumped by f(x+1). Because evaluation is central to the concept of a function, I would be very hesitant to suggest anything that obscures it.

[HallsofIvy]

Strictly speaking, if "f" is a function, then "f(x)" is the (numerical) value when f is applied to the value x.

[roger]

Dear HallsofIvy,

what if the equation is 3x^2 + 5x +9 = Y

and It says f(x+3)

Does it mean replace all of the x in the equation by x+3 ?

I'm trying to understand the difference between af(x) and f(ax) ?


Thankyou for any advice


roger

[shmoe]

if f(x)=3x^2 + 5x +9

f(junk) tells you to take your expression for f(x) and replace every x you see with "junk", whatever the junk is. Don't hold out on the brackets, they are inexpensive.

f(x+3)=3(x+3)^2 + 5(x+3) +9

Expand out if you wish to simplify.

f(ax) is the function applied to ax, so

f(ax)=3(ax)^{2}+5(ax)+9=3a^{2}x^{2}+5ax+9

af(x) is the function applied to x, then the whole thing multiplied by a:

af(x)=a(3x^{2} + 5x +9)=3ax^{2}+5ax+9a


Using f(x) as above can you find f(y)? f(x+2y)? f(x^2)? (f(x))^2?

[HallsofIvy]

Nicely done, Shmoe

[roger]

if f(x)=3x^2 + 5x +9

f(junk) tells you to take your expression for f(x) and replace every x you see with "junk", whatever the junk is. Don't hold out on the brackets, they are inexpensive.

f(x+3)=3(x+3)^2 + 5(x+3) +9

Expand out if you wish to simplify.

f(ax) is the function applied to ax, so

f(ax)=3(ax)^{2}+5(ax)+9=3a^{2}x^{2}+5ax+9

af(x) is the function applied to x, then the whole thing multiplied by a:

af(x)=a(3x^{2} + 5x +9)=3ax^{2}+5ax+9a


Using f(x) as above can you find f(y)? f(x+2y)? f(x^2)? (f(x))^2?

Dear Shmoe,

I've had a go, is this correct :

f(y) = 3y^2 + 5y + 9

f(x+2y) = 3(x+2y)^2 +5(x+2y) + 9

f(x^2) = 3x^4 + 5x^2 + 9

(f(x))^2 = (3x^2+5x+9)^2

But what does it actually mean by f(y) ?
Does it mean y is a function of x ?


thanx


roger

[arildno]

All your go's are correct.
Without any other specification, inserting "y" in the x-place, (that is, in your first equation),
you've simply changed the variable name from "x" to "y" (just a notational change).

[MiGUi]

> f(x+2y) = 3(x+2y)^2 +5(x+2y) + 9

In this example, you say that f depends on x and also on y, so it is a function of x and y and it is written as f(x,y). The trick of shmoe is valid, but don't forget that the information given between the brackets is the dependance of the function.