Functions And Graphing

I.  Homework

II.  Definition of a function

A function is a rule that assigns every element from a set (called the domain) to a unique element of a set (called the range)

Examples

A)  Let the domain be US citizens and the range be the set of all fathers.  Let the rule of the function send each person to his or her fathers.  

B)  f(x) = x²  

C)  f(x) = 3x + 1

D)  f(x) = 1/x  (the domain is R - 0)

E)  x² + y²  = 1  (not a function since for x = 0, y can be 1 or -1.

III.  Vertical Line Test

If every vertical line passes through the graph at most once, then the graph is the graph of a function.

Examples will be given in class

IV.  Finding the Domain

To find the domain we follow the three basic principals:

1)  The domain of a polynomial is R

2)  The domain of a rational function p(x)/q(x) all real numbers x such that q(x) is not 0.

3)  The domain of a square root function is all real numbers such that the function inside the square root is non-negative.

4)  For a word problem the domain is the set of all x values such that the problems makes sense.

Examples:  

The domain of (x + 1)/(x - 1) is all real numbers except where x = 1.

The class will do other examples.

V.  The Difference Quotient and Function Evaluation

If f is an algebraic function we can write without the variables as in the following example:

If f(x) = x² - 2x

We can write

f = (  )² - 2(  )  

This more suggestively shows how to deal with non x inputs.  For example

f(x - 1) = (x - 1)² - 2(x - 1)

The Difference Quotient

We define the difference quotient for a function f by

[f(x + h) - f(x)]/h

For the above example we have

f(x + h) = (x + h)² - 2(x + h) = x² + 2xh + h² -2x - 2h  

f(x) = (x)² - 2(x)

so that

[f(x + h) - f(x)]/h = [x² + 2xh + h² -2x - 2h - (x² - 2x)]/h

= [2xh + h² - 2h]/h

= [h(2x + h - 2)]/h

= 2x + h - 2

VI.  Finding the domain and range of a function from its graph.

We will be using the graphing calculator to find the domain and range of functions.  In general, the domain will be the set of all x values that has corresponding points on the graph.  We note that if there is an asymptote (shown as a vertical line on the TI 85) we do not include that x value in the domain.  To find the range, we seek the top and bottom of the graph.  The range will be all points from the top to the bottom (minus the breaks in the graph).  This will be demonstrated in class.

VII.  Zeros of Functions and the x-intercept Method

To find the x-intercepts of a complicated function, we can use the TI 85 to view the graph, then use more math root to find the x-intercepts.

We will try this on

y = x³ - 4x² + x + 2

And others.  

If we want to find the intercept of two graphs, we can set them equal to each other and then subtract to make the left hand side zero.  Then set the right hand side equal to y and find the zeros.

Example:  

Find the intercept of the graphs:

y = 2x³ - 4 and

y = x⁴ - x² 

We will do this on the TI85.

VIII. Solving Inequalities Graphically 

To solve an inequality graphically we first put 0 on the right hand side and f(x) on the left hand side.  Then we use the x-intercept method to find the zeros.  If the inequality is a < we include the part of the graph below the x axis.  If the inequality is a > we include the part of the graph above the x axis.  We will demonstrate examples on the TI 85 calculator.  

IX.  Word Problems

We will try number 56 and 58 in class.