
Definition of a Function
Definition
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

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.

f (x) = x^{2}

f (x) = 3x + 1

f (x) = 1/x (the domain is
R  {0})

x^{2} + y^{2} = 1
(not a function since
for
x =
0
y can be 1 or 1.

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

Finding the Domain
To find the domain of a function, we follow the three basic principals:

The domain of a polynomial is R
The domain of a rational function
p(x)/q(x)
is the set of all real numbers x such
that
The domain of a square root function is all real numbers such that
the function inside the square root is nonnegative.
For example, to find the domain of
we just set
4  2x
> 0
or
x <
2
so that the domain of f(x)
is
{x  x
< 2}
For a word problem the domain is the set of all x values such that
the problems makes sense.
Examples:
The domain of
is all real numbers except where x = 1.

Function Evaluation
If f is an algebraic function we can write without the variables as in the
following example:
If
f(x) =
x^{2}  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
Difference Quotient

Example
Let
f(x) =
x^{2}  2x
Then
f(x +
h) = (x + h)^{2}  2(x + h) = x^{2} +
2xh +
h^{2} 2x  2h
and
f(x) =
(x)^{2}  2(x)
so that

Finding the domain and range of a function from its graph.
We often use 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).

Zeros of Functions and the xintercept Method
To find the xintercepts of a complicated function, we can use the TI 85
to view the graph, then use
more math root
to find the xintercepts.
Exercise
Find the roots of
y = x^{3} 
4x^{2} + x + 2
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 = 2x^{3}  4
and
y = x^{4} 
x^{2}
Solution:
We form the new function
f(x)
= (2x^{3}  4)  (x^{4}  x^{2})
and use the root feature of our calculator to get
x =
1.52 or x = 2

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 xintercept 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.
Example:
Give and approximate solution of
3x^{5}
 14x^{2} < x  4
Solution:
First, bring everything to the left hand side
3x^{5}
 14x^{2}  x + 4 < 0
The graph shows that the roots lie at:
x =
0.56, x = 0.51, and x
= 1.64
The points lie below the xaxis to the left of 0.56 and between 0.51 and
1.64. Hence the solution in interval notation is