Lesson FUNCTIONS

Source code of 'FUNCTIONS'

This Lesson (FUNCTIONS) was created by by Theo(13342)

: View Source, Show
About Theo:

This lesson will give a brief overview of functions.

Inverse functions are not covered here.
There will be a separate lesson on them that will be titled INVERSE FUNCTIONS.

Most of what is covered here comes from the following web address:

<a href = "http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut30_function.htm" target = "_blank">http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut30_function.htm</a>

If you want a detailed tutorial on functions and inverse functions with lots of examples and practice problems, go there.

There are lots of other tutorials out there.

Here's a list of some of them.

REFERENCES
<a href = "http://www.purplemath.com/modules/fcns.htm" target = "_blank">http://www.purplemath.com/modules/fcns.htm</a>
<a href = "http://www.themathpage.com/aprecalc/functions.htm" target = "_blank">http://www.themathpage.com/aprecalc/functions.htm</a>
<a href = "http://www.themathpage.com/aprecalc/functions.htm" target = "_blank">http://www.themathpage.com/aprecalc/functions.htm</a>
<a href = "http://tutorial.math.lamar.edu/Classes/Alg/Graphing_Functions.aspx" target = "_blank">http://tutorial.math.lamar.edu/Classes/Alg/Graphing_Functions.aspx</a>
<a href = "http://www.purplemath.com/modules/fcnnot.htm" target = "_blank">http://www.purplemath.com/modules/fcnnot.htm</a>
<a href = "http://regentsprep.org/Regents/math/algtrig/ATP5/EvaluatingFunctions.htm" target = "_blank">http://regentsprep.org/Regents/math/algtrig/ATP5/EvaluatingFunctions.htm</a>

DEFINITION OF AN ORDERED PAIR

An ordered pair would be a set of (x,y) values where x is the independent variable, and y is the dependent variable.
This means that if you pick any x value in the set of all possible x values, the y value will be dependent on that x value in the equation or rule that associates that x value with the y value.

let an equation be y = {{{x^2}}}
For every value of x, you have a value of y that is dependent on that value of x in the equation.
if x = 5, y = 25
if x = 6, y = 36, etc.
The ordered pair in this relation or function would be
(x,y) = (5,25)
(x,y) = (6,36), etc.

Ordered pairs are useful in graphing, where you have an x-axis (horizontal line), and a y-axis (vertical line). 
The x-axis is a horizontal line drawn where the value of x = 0.
The y-axis is a vertical line drawn where the value of y = 0. 
The point (x,y) on the graph is related to the value of x and the value of y in the following manner:

let x = 3
let y = 7
the point (3,7) is 3 units of measure to the right of x = 0, and 7 units of measure above y = 0.

let x = -2
let y = -3
the point (-2,-3) is 2 units of measure to the left of x = 0, and 3 units of measure below y = 0

an equation for a line that passes through these 2 points would be:  {{{y = 2*x+1}}}
Both these points will be on that line.

The ordered pair associated with that equation would be:
(x,y) = (x,(2*x+1))
The x value is x.
The y value is 2*x+1

a graph of that equation would look like:

{{{graph(600,300,-4,4,-8,8,2*x+1)}}}

DEFINITION OF A RELATION

A relation is a set of ordered pairs where the first components of the ordered pairs are the input values and the second components are the output values

In a relation, you can have multiple y values for each x value.
Example:

y = +/- {{{sqrt(x)}}}
for every value of x, y can be +/- {{{sqrt(x)}}}
assume x = 25, then:
y can be + 5 and y can be - 5.
both values for y are good because:
{{{(-5)^2}}} = 25 and {{{(+5)^2}}} = 25

ordered pairs in this relation when x = 25 would be:
(25,5)
(25,-5)

DEFINITION OF A FUNCTION

A function is a relation that assigns to each input number EXACTLY ONE output value for each value of x in the domain.

While a relation allows multiple y values for each x, a function requires that there can only be one.  If any one value in the domain of the equation has more than one y value associated with it, then the equation is a relation and not a function.

DOMAIN

The domain is the set of all input values to which the rule or equation applies.  These input values are called your independent variables. These are the values that correspond to the first component of the ordered pairs it is associated with.  The value of x in the domain must associate with a corresponding value of y in the range.

RANGE

The range is the set of all output values.  These are called your dependent variables. These are the values that correspond to the second component in the ordered pairs it is associated with.  For every value of x in the domain, there will be at least one value of y in the range.

Please note that values of x or y imply real values only.  
infinity is not a real value since there is not one value that can represent it.  You can approach it but you will never reach it.
square root of a negative number is not a real value since it is an imaginary value which is not part of the set of real values.

FUNCTIONAL NOTATION

You have an equation, like y = {{{x^2 + 2}}}
If you replace y with f(x), then the equation looks like f(x) = {{{x^2 + 2}}}
You can say y = f(x) = {{{x^2 + 2}}}, or simply f(x) = {{{x^2 + 2}}} or simply y = {{{x^2 + 2}}}
What you are saying is that y is a function of x where the value of y is dependent on the value of x in the equation {{{x^2 + 2}}}.
x is the independent variable
y is the dependent variable

example:
What is the value of y when x is 3?
Since y = f(x) = {{{x^2 + 2}}}, then when the value of x = 3,
y = f(3) = {{{(3)^2 + 2}}} = 9 + 2 = 11.
the ordered pair would be (3,11).

Functional Notation can use any letter, not just f.
f(x) = {{{x^2+1)}}}
g(x) = {{{x^2+1)}}}
h(x) = {{{x^2+1)}}}
i(x) = {{{x^2+1)}}}
All of these letters can represent a function of x which is equal to {{{x^2+1)}}} in this case.

Functions can be functions of any independent variable.
f(x) is a function of x.
f(t) is a function of t.
A(r) is a function of the radius of a circle.  The A stands for Area.  Your equation would be:
{{{A(r) = pi*r^2}}}
You can say {{{A(r) = pi*r^2}}}
Or you can say {{{y = A(r) = pi*r^2}}}
Or you can say {{{y = pi*r^2}}}
The same equation or relationship between the output variable and the input variable apply.
Ordered pairs could look like any of the following:
(r,y)
(r,A(r))
(r,{{{pi*r^2}}})
(x,y) where x = 4 and y = A(r) = {{{pi*r^2}}}
They are all equivalent.

OPERATIONS ON FUNCTIONS

Functions can be added, subtracted, multiplied, and divided.
You can also take the function of a function.

DEFINITION OF FUNCTION ADDITION

{{{(f + g)(x) = f(x) + g(x)}}}

Example:
let f(x) = 2*(x+1)
let g(x) = (x+1)
{{{(f + g)(x) = f(x) + g(x) = 2*(x+1) + (x+1) = 3*(x+1)}}}

DEFINITION OF FUNCTION SUBTRACTION

{{{(f - g)(x) = f(x) - g(x)}}}

Example:
let f(x) = 2*(x+1)
let g(x) = (x+1)
{{{(f - g)(x) = f(x) - g(x) = 2*(x+1) - (x+1) = (x+1)}}}

DEFINITION OF FUNCTION MULTIPLICATION

{{{(f * g)(x) = f(x) * g(x)}}}

Example:
let f(x) = 2*(x+1)
let g(x) = (x+1)
{{{(f * g)(x) = f(x) * g(x) = 2*(x+1) * (x+1) = 2*(x+1)^2}}}

DEFINITION OF FUNCTION DIVISION

{{{(f / g)(x) = f(x) / g(x)}}}

Example:
let f(x) = 2*(x+1)
let g(x) = (x+1)
{{{(f/g)(x) = f(x) / g(x) = 2*(x+1) / (x+1) = 2}}}

DEFINITION OF COMPOSITE FUNCTION

When you take the function of a function, then you are dealing with a composite function.
The formula will look like:
{{{(f o g)(x) = f(g(x))}}} where the o in the middle is really a small circle that I am incapable of reproducing, so I used the small letter o to represent it.

{{{(f o g)(x)}}} means {{{f(g(x))}}} which means f is a function of g which is a function of x.

AN EXAMPLE OF {{{(f o g)(x)}}}

let {{{g(x) = x^2}}}
let {{{f(x) = x^3}}}
then:
{{{(f o g)(x) = f(g(x)) = f(x^2) = (x^2)^3 = x^6}}}

You are doing what you probably already know how to do with functions, which is replace the x with whatever value you are dealing with.

{{{f(x) = x^3}}}

what is f(2)?
you replace the x with 2 and solve:
{{{f(2) = (2)^3 = 2*2*2 = 8}}}

what is {{{f(x^2)}}}?
you replace the x with {{{x^2}}} and solve:
{{{f(x^2) = (x^2)^3 = x^2*x^2*x^2 = x^6}}}

what is {{{f(g(x))}}}?
you replace the x with {{{g(x)}}} and solve:
{{{f(g(x)) = (g(x))^3}}}
since g(x) = {{{x^2}}}, you replace g(x) with {{{x^2}}} to get:
{{{f(g(x)) = (g(x))^3 = (x^2)^3 = x^2*x^2*x^2 = x^6}}}

GRAPH OF A FUNCTION VERSUS GRAPH OF A RELATION

The graph of an equation can help you to determine if you are dealing with a function or a relation.
A function has one and only one value of y for each x.

VERTICAL LINE TEST

If you draw a vertical line at any value of x in the domain of the equation, that vertical line will cross the equation at one and only one point.  If it crosses at more than one point, then you do not have a function.

GRAPH OF A FUNCTION

Take the function {{{f(x) = x^3}}}
The domain is all real values of x.
The range is all real values of y which is the same as all real values of f(x).
The graph looks like the following:

{{{graph(600,300,-10,10,-100,100,x^3)}}}

This equation is a function because you can draw a vertical line through any value of x in the domain and it will cross the graph of the equation only once.

GRAPH OF A RELATION

Take the relation {{{y = f(x) = sqrt(x)}}}
The domain is all positive real values of x including 0.  Negative values of x are not possible because you can't take the square root of a negative number and get a real number as an answer.
The range is all real values of y which can be all real numbers since the square root of x can be negative or positive, even though x itself has to be positive.
The graph looks like the following:

{{{graph(600,300,-10,10,-5,5,x^(1/2),-x^(1/2))}}}

This equation is a relation because you can find at least one value of x in the domain of the equation where there is more than one value of y for that x.  All you need is one.  Here you have many.