10.6
Introduction To Functions
Learning Objectives:
1. Identify relations, domains, and ranges.
2. Identify functions.
3. Use the vertical line test.
4. Use function notation.
5. Key Vocabulary: relation, domain, range, function, vertical line test, function notation.
A. Identifying Relations, Domains and Ranges:
Definition:
Relation—is a set of ordered pairs.
Domain of the relation—is the set of all possible x-values.
Range of the relation—is the set of all possible y-values.
Types of Functions
1. Linear Function:
y = mx + b ;
Domain: all real numbers;
Range: all real numbers
2.
Quadratic Function:
y = ax 2 + bx + c ;
Domain: all real numbers
3.
Rational Function: y =
P
;
Q
all real numbers except Q = 0
Domain:
Range: all real numbers except y = 0
Example 1. Find the domain and range:
T = {(6, − 4 ), (4, − 6), (− 1, 3)}
B. Identifying Functions
Function—is a set of ordered pairs in which each domain value has exactly one range value; that is, no
two different ordered pairs have the same first coordinate.
Function Notation:
f ( x ) read “ f of x” or “f evaluate at x”
Example 2.
1.
Determine whether the relations are functions:
T = {(6, − 4 ), (4, − 6), (− 1, 3)}
2.
T = {(6, − 4 ), (4, − 6), (− 1, 3), (4, 3)}
C. Using the Vertical Line Test
Vertical Line Test—if a vertical line can be drawn so that it intersects a graph more than once, then
graph is not the graph of a function.
Example 3. Determine whether the graph is that of a function.
1.
2.
y
y
5
5
–5
5
x
−5
D.
Evaluating Functions
Example 4.
Let f ( x ) = x 3 − 3x 2 + 2 x − 4 , find
1.
f (− 2 )
2.
f (2) − f (− 1)
–5
5
–5
x