Transcript Document

1.2
Basics of
Functions and
Their Graphs
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Relations
The test grades of 22 students in Mrs. Smith’s Algebra
class are shown in the table below. The table indicates a
correspondence between a grade and the number of
students receiving that grade.
Grade
# of Students
A
5
B
7
C
8
D
1
F
1
We can write this correspondence
as a set of ordered pairs:
{(A, 5), (B, 7), (C, 8), (D, 1), (F, 1)}.
The mathematical term for a set
of ordered pairs is a relation.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Blitzer, College Algebra: An Early Functions Approach, 2ed
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Relations
Definition of a Relation
A relation is any set of ordered pairs. The set of all first
components of the ordered pairs is called the domain of
the relation and the set of all second components is called
the range of the relation.
Example:
Find the domain and range of the relation:
{(A, 5), (B, 7), (C, 8), (D, 1), (F, 1)}.
Domain: {A, B, C, D, F}
Range: {5, 7, 8, 1}
The 1 is only listed once.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Blitzer, College Algebra: An Early Functions Approach, 2ed
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Functions
A relation in which each member of the domain
corresponds to exactly one member of the range is a
function.
# of Students
Grade
Grade
# of Students
A
5
B
7
C
8
A
B
C
D
F
D
1
Domain
F
1
5
7
8
1
Range
Grades corresponding to number of students.
This relation is a function.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Blitzer, College Algebra: An Early Functions Approach, 2ed
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Functions
# of Students
Grade
# of Students
A
5
B
7
C
8
D
1
F
1
5
7
8
1
Domain
Grade
A
B
C
D
F
Range
Number of students corresponding to grades.
This relation is NOT a function.
Definition of a Function
A function is a correspondence from a first set, called the domain,
to a second set, called the range, such that each element in the
domain corresponds to exactly one element in the range.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Blitzer, College Algebra: An Early Functions Approach, 2ed
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Functions
Example:
Determine whether the relation is a function:
{(3, 2), (0, 8), (2, 3), (3, 4), (4, 4)}.
This is not a function
because 3 corresponds
to both 2 and 4.
3
0
2
4
Domain
2
8
3
4
Range
A function can have two different first components with the
same second component. By contrast, a relation is not a function
when two different ordered pairs have the same first component
and different second components
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Blitzer, College Algebra: An Early Functions Approach, 2ed
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Functions as Equations
Functions are usually given in terms of equations rather
than as sets of ordered pairs.
y is a function of x.
y = 3x + 5
For each value of x, there is only one value of y.
The variable x is called the independent variable
because it can be assigned any value from the domain.
The variable y is called the dependent variable
because its value depends on x.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Blitzer, College Algebra: An Early Functions Approach, 2ed
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Functions as Equations
Example:
Determine whether the equation defines y as a function of x.
x2 + y2 = 9
Solve the equation for y in terms of x
x2 + y2 = 9
x2 + y2 – x2 = 9 – x2
y2 = 9 – x2
y   9  x2
Solve for y.
Simplify.
Apply the square root property.
Since two or more values of y can be obtained for a
given x, the equation is not a function.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Blitzer, College Algebra: An Early Functions Approach, 2ed
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Function Notation
If an equation in x and y gives one and only one value of y
for each value of x then the variable y is a function of the
variable x.
The special notation f(x), read “f of x” or “f at x,”
represents the value of the function at the number x.
Input
x
Output
f(x)
Equation
This is read as
f(x) = 3x2 + 8.
f(x) = 3x2 + 8
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Blitzer, College Algebra: An Early Functions Approach, 2ed
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Evaluating a Function
Example:
Evaluate the function for f(– 2).
f(x) = 3x2 + 8
f(x) = 3(– 2)2 + 8
Substitute – 2 for x in the equation.
= 3(4) + 8
Square – 2.
= 12+ 8
Multiply.
= 12+ 8
Add.
= 20
Simplify.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Blitzer, College Algebra: An Early Functions Approach, 2ed
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Graphs of Functions
The graph of a function is the graph of its ordered pairs.
f(x) = – x + 3
Example:
Graph the function f(x) = x + 3.
y
(2, 5)
4
x
f(x) = – x + 3
(x, y)
1
y = (1) + 3 = 4
(1, 4)
2
y = (2) + 3 = 5
(2, 5)
1
y = (– 1) + 3 = 2
(1, 2)
3
(1, 2)
(1, 4)
2
1
x
4
3 2 1
1
2
3
4
2
3
4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Blitzer, College Algebra: An Early Functions Approach, 2ed
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The Vertical Line Test
The Vertical Line Test for Functions
If any vertical line intersects a graph in more than one
point, the graph does not define y as a function of x.
FUNCTION
FUNCTION
NOT A
FUNCTION
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Blitzer, College Algebra: An Early Functions Approach, 2ed
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