Lesson BASICS - Functions

Source code of 'BASICS - Functions'

This Lesson (BASICS - Functions) was created by by longjonsilver(2297)

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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

<A HREF=Plot-a-graph-of-functions.solver>Draw a graph of any function</A>, or <A HREF=Plot-A-Graph-With-Domain.solver>draw a graph of a function and show domain</A>.

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 <A HREF=Plot-Inverse-Function.solver>Plot a function and its inverse on the same graph</A>, 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^(-1)(y) = y+4}}}. But we tend to write functions in terms of x, that is all, so we tend to re-write this as {{{f^(-1)(x) = x+4}}}

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^2}}} curves             {{{graph( 200, 200, -10, 10, -10, 10, x^2)}}}
Learn the shape of y={{{x^3}}} curves             {{{graph( 200, 200, -10, 10, -10, 10, x^3)}}}
Learn the shape of y={{{1/x}}} curves             {{{graph( 200, 200, -10, 10, -10, 10, 1/x)}}}
Learn the shape of y={{{(x+a)/(x+b)}}} curves     {{{graph( 200, 200, -10, 10, -10, 10, (x+1)/(x+2))}}}
Learn the shape of y={{{a^x}}} curves             {{{graph( 200, 200, -10, 10, -10, 10, 2^x )}}}
Learn the shape of y={{{log(x)}}} curves          {{{graph( 200, 200, -10, 10, -10, 10, ln( x ))}}}
Learn the shape of y={{{1/x^2}}} curves           {{{graph( 200, 200, -10, 10, -10, 10, 1/x^2)}}}