Functions

[Okami Wolfen]

What is a function?
How do I tell if a problem is a function?
What is the inverse of a function?
What the hell does this mean?

If f(x) = 2x and g(x) = x-4, what is the value of f[g(3)]?

Help me? Plz?

[southern belle]

f(x) = 2x is an example of a function... basically it's where, in a set problem, the function(s) (f and g in the above problem) can be substituted by the value on the other side of the = (2x and x-4).

So, f[g(3)] would be 2x[(x-4)(3)]. I imagine they probably want you to simplify it further from there.

Edit: I dug out my algebra book, and this is what the glossary says under function:
A function consists of two sets, the domain and the range, together with a rule which assigns each member of the domain to exactly one member of the range. Each member of the range must be assigned to at least one member of the domain.

Uhh... hope that helps. XD

[DarkZeroHyrule]

A function is basically a "y=" equation except the "y" is replaced with an "f(x)" which means "the value of 'f' (the line) at 'x' (the numbers they give you)". The main thing to remember here is that every x value has exactly one y value, or else it is not a real function and cannot be graphed as such. Like southern belle said, f(g(x)) is when you plug the function g(x), a separate function, into f(x). Taking you problem:

f(x)= 2x g(x)= x-4

f(g(x)) = f(x-4) <---- I just took the function g(x) and changed it to what it was equal to, which in this case was x-4.

f(x-4) = 2(x-4) <---- I now took what was in the parentheses (x-4) and put it wherever x was in f(x). Now, since the format is f(g(x)) = 2(x-4) and they want f(g(3)), then x must equal 3. So, plug 3 in for x and your answer will come out like toast... or a bagel... or croissant... or whatever pastry of your choosing.

Now, an inverse function is when you switch the y values [f(x) = y] and the x values [what is in the parentheses in f(x)]. I'll give you a table for f(x) = 2x.

X values | 0 | 1 | 2 | 3 | 4 | 5 |
Y values | 0 | 2 | 4 | 6 | 8 | 10 |

Now, when you switch the x and y...

X values | 0 | 2 | 4 | 6 | 8 | 10 |
Y values | 0 | 1 | 2 | 3 | 4 | 5 |

This is a new function. If you have a graphing calculator, great! If not, graphing paper works well. If not, search google for Graphmatica, an excellent program that graphs functions for you on your computer (yay). Technology is awesome, no?

Anyway, if you plot those points, you'll see they move away from each other. Now, plot these points.

X values| 0 | 1 | 2 | 3 | 4 | 5
Y values| 0 | 1 | 2 | 3 | 4 | 5

Connect these new dots only. This is the line y = x which can also be called the function f(x) = x. Now, connect the dots for each table separately, so the first set has one line and the second set has another. As you can probably see, they are reflected across this magical function of f(x) = x. That is the main characteristic of an inverse function. Inverse functions mirror each other across the line y = x. This is because the x values of one function happen to be the y values of another and vice-versa. Now, to prove that this is true algebraically, that is a lesson for another day. It involves knowing functions, how inverse functions generally work, and the concept of plugging g(x) into f(x) and the other way around, which is all explained above by southern belle and myself. If you need any more math related help, feel free to ask.

I never thought I'd miss algebra. Math was easier then...

[Dog Dog 123]

Eh, a function would just mean that only one X for each Y. Usually a kid would be taught to check for functions with the the vertical line test. In other words, take each point, follow the line, and see if there are any other points on that line.

Inverse function, I'm not entirely sure about, but it would probably just be X and Y switched around in the above.

f(x) is simply another way to write Y. You know how all equations have been y=4x? Now it's simply become f(x)=4x. See?

Now, if it's a number in the parenthesis, like f(4)=4x, you'd solve the equation for 4.

Now, if you're given a G and an F, you might be told to add those functions, or subtract, or do the annoying one that you have up above.

If f(x) = 2x and g(x) = x-4, what is the value of f[g(3)]?

Alrighty, so try moving through it one step at a time. You'd be solving g(3) first. g(3)=x-4. 3-4=-1. Now that you've solved the first part of that equation, you'd replace the g(3) with a -1. So now your equation becomes f(-1) because you've replaced the g(3) with the solution of that piece. So now you solve f(-1). f(-1)=2x. 2(-1)=-2. Therefore f(g(3))=-2. Unless my math is wrong, which it probably is.

Or, if your teacher doesn't like that way and wants you to transfer it all out into one equation, you'd replace the X of teh first equation with the second equation.

f(g(3))=2(x-4)=2(3-4)

And then probably solved the rest of the way from there.

Does that make sense? I'm a bad explainer...


EDIT: And I'm too slow...