Lesson BASICS - Functions

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This Lesson (BASICS - Functions) was created by by longjonsilver(2297) About Me : View Source, Show
About longjonsilver: I have a new job in September, teaching
As an initial entry to the world of functions, just consider them to be a more formal representation of the equation y=... etc.
So, f(x) = 3x-1 is, at first glance, just another, more professional way of saying y=3x-1.
How to read a function

f(x) = 3x-1 means whatever the "x" value is in the bracket of f(x), that value must be put into the function

eg f(2) --> 3(2)-1 --> 6-1 --> 5
eg f(-1) -> 3(-1)-1 -> -3-1 -> -4
f(2y) --> 3(2y)-1 --> 6y-1

etc
How to plot a graph of a function

Draw a graph of any function, or draw a graph of a function and show domain.

Composite Functions

If f(x)=2x and g(x)=1-x, find ff(x), gg(x), fg(x) and gf(x)

To work these out, just write whatever is in the bracket into the function, so
1. ff(x) become f(2x) which become 2(2x) --> 4x

2. gg(x) becomes g(1-x) which becomes 1-(1-x) --> x

3. fg(x) becomes f(1-x) which becomes 2(1-x)

4. gf(x) becomes g(2x) which becomes 1-2x

As you can see, fg(x) and gf(x) are not necessarily the same thing, so the combining of functions is not like numbers, where 2*3 is the same as 3*2, for example.

Inverse Functions
You can Plot a function and its inverse on the same graph, a nifty little solver.
An inverse function, is basically the reverse function. By this I mean, starting with f(x)=x-4. If we picked a value of x, say 5...this gives an answer of 5-4, which is 1. The inverse function would use the 1 and arrive back at the 5.
How do we find this inverse? Well, basically, we just re-write the function y=x-4 as x= whatever.

In this example, we would have x=y+4.

So, the inverse function is f%5E%28-1%29%28y%29+=+y%2B4. But we tend to write functions in terms of x, that is all, so we tend to re-write this as f%5E%28-1%29%28x%29+=+x%2B4

DOMAIN and RANGE

Again, trying to keep it simple:

DOMAIN is all the possible x-values you are allowed to put into the function
RANGE is all the y-values you get out of the function, when you put all the possible x-values in.

The best way to find the domain and range is to know the shape of a few basic graphs:

Learn what y=ax+b looks like
Learn the shape of y=x%5E2 curves graph%28+200%2C+200%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E2%29
Learn the shape of y=x%5E3 curves graph%28+200%2C+200%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E3%29
Learn the shape of y=1%2Fx curves graph%28+200%2C+200%2C+-10%2C+10%2C+-10%2C+10%2C+1%2Fx%29
Learn the shape of y=%28x%2Ba%29%2F%28x%2Bb%29 curves graph%28+200%2C+200%2C+-10%2C+10%2C+-10%2C+10%2C+%28x%2B1%29%2F%28x%2B2%29%29
Learn the shape of y=a%5Ex curves graph%28+200%2C+200%2C+-10%2C+10%2C+-10%2C+10%2C+2%5Ex+%29
Learn the shape of y=log%28x%29 curves graph%28+200%2C+200%2C+-10%2C+10%2C+-10%2C+10%2C+ln%28+x+%29%29
Learn the shape of y=1%2Fx%5E2 curves graph%28+200%2C+200%2C+-10%2C+10%2C+-10%2C+10%2C+1%2Fx%5E2%29
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