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...
