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MATH 1100
SECTION 4.1 Notes
What is a Function? – Text Pages 209-216
Introduction to Functions – Function All Around Us:
Read the text for a well written introduction to function!!
Function:
A function is a rule that assigns to each element x in a set A
exactly one element, called f(x), in a set B.





f(x):
Say: “f of x”
Called: “the value of f at x”
The set A: is called the domain of the function
The range of a function is the set of all possible values of f(x) as x
varies throughout the domain,  f ( x) : x  A .
The variable that represents an element in the domain of the
function is called the independent variable, and the variable that
represents an element in the range is called the dependent
variable.
If the domain of the function is not specifically stated, such as
1<x<5, then the domain is the set of all real numbers for which the
expression is defined as a real number.
Guidelines for Determining the Domain:
 The domain of all linear functions is the set of all real
numbers, .
 If there is a variable in the denominator, the domain cannot
contain any value that makes the denominator equal to zero.
 If there is a variable under a square root symbol, the domain
cannot contain any value that creates the square root of a
negative number.
Example 1: A function can be represented by a set of points, x, f x  .
Determine if each set of points represents a function:
(a)
3,  1, 4, 0,  2, 6, 4, 7
(b)
3,
 1, 4, 0,  2, 6, 5, 2
(c)
1, 5, 2, 5, 3, 5, 4, 5, 5, 5
Example 2: The squaring function assigns to each real number x its
square, x2. It is defined by,
f x   x 2 .
a)
Evaluate:
f 3 
f  2 
f
b)
 5
Find:
The domain of f =
The range of f =
(c)
Draw a machine diagram for f:
Example 3:
(a)
f 4 
(b)
f  2 
(c)
f a  
f x   2 x  5
Example 4:
f x   4 x 2  6 x  10
(a)
f 1 
(b)
f a  
(c)
f  a  
(d)
f a  h 
(e)
f a  h   f a 

h
where h  0
Example 5: Find the domain of each function:
(a)
f x   5 x  1
Domain =
(b)
f x  
Domain =
(c)
f x  
1
2x  6
Domain =
(d)
f x  
2x  1
x7
Domain =
(e)
f x  
x5
x2 1
Domain =
x
Example 6: The function in this example is called a Piecewise Defined
Function because it is defined differently for different pieces
of its domain.
3x
, x0

f x    x  1
, 0 x2

2
x  2 , x  2
(a)
f  5 
(b)
f 0 
(c)
f 2 
(d)
f 10 