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Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Section 2.1
Functions and Their
Representations
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Objectives
• Basic Concepts
• Representations of a Function
• Definition of a Function
• Identifying a Function
• Tables, Graphs and Calculators (Optional)
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
The notation y = f(x) is called function notation. The input is x, the output is y, and the name of the function is f.
Name
y = f(x)
Output Input
FUNCTION NOTATION
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The variable y is called the dependent variable and the variable x is called the independent variable. The expression f(4) = 28 is read “f of 4 equals 28” and indicates that f outputs 28 when the input is 4. A function computes exactly one output for each valid input. The letters f, g, and h, are often used to denote names of functions.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Representations of a Function
Verbal Representation (Words)
Numerical Representation (Table of values)
Symbolic Representation (Formula)
Graphical Representation (Graph)
Diagrammatic Representation (Diagram)
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Example
Evaluate f(x) at the given value of x.f(x) = 5x – 3 x = −4
Solutionf(−4) = 5(−4) – 3
= −20 – 3= −23
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Example
Let a function f compute a sales tax of 6% on a purchase of x dollars. Use the given representation to evaluate f(3).SolutionVerbal Representation Multiply a purchase of x dollars by 0.06 to obtain a sales tax of y dollars.
Numerical Representation x f(x)
$1.00 $0.06
$2.00 $0.12
$3.00 $0.18
$4.00 $0.24
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Example (cont)
Let a function f compute a sales tax of 6% on a purchase of x dollars. Use the given representation to evaluate f(3).SolutionSymbolic Representation f(x) = 0.06x
Graphical Representation
X
Y
1 2 3 4 5 6
0.1
0.2
0.3
0.4
0.5
0.6
0
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example (cont)
Let a function f compute a sales tax of 6% on a purchase of x dollars. Use the given representation to evaluate f(3).SolutionDiagrammatic Representation
1 ●
2 ●
3 ●
4 ●
● 0.06
● 0.12
● 0.18
● 0.24
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Let function f square the input x and then add 3 to obtain the output y. a.Write a formula for f. b.Make a table of values for f. Use x = −2, −1, 0, 1, 2.c.Sketch a graph of f. Solutiona. Formula If we square x and then add 3, we obtain x2 + 3. Thus the formula is f(x) = x2 + 3.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example (cont)
b. Make a table of values for f. Use x = −2, −1, 0, 1, 2.c. Sketch the graph.Solution
x f(x)
−2 7
−1 4
0 3
1 4
2 7
X
Y
-5 -4 -3 -2 -1 1 2 3 4 5
-2
-1
1
2
3
4
5
6
7
8
0
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
A function receives an input x and produces exactly one output y, which can be expressed as an ordered pair:
(x, y)
Input Output
Definition of a Function
A relation is a set of ordered pairs, and a function is a special type of relation.
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A function f is a set of ordered pairs (x, y), where each x-value corresponds to exactly one y-value.
Function
The domain of f is the set of all x-values, and the range of f is the set of all y-values.
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Example
Use the graph to find the function’s domain and range.
Domain
Range
3 3D x
0 3R y
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Example
Use the graph to find the function’s domain and range.
all real numbersD
4R y
Domain
Range
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example
Use f(x) to find the domain of f.
a. f(x) = 3x b.
Solutiona. Because we can multiply a real number x by 3, f(x) = 3x is defined for all real numbers. Thus the domain of f includes all real numbers.
b. Because we cannot divide by 0, the input x = 4 is not valid. The domain of f includes all real numbers except 4, or x ≠ 4.
1
4f x
x
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Slide 17
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Example
Determine whether the table represents a function.
x f(x)
2 −6
3 4
4 2
3 −1
1 0
The table does not represent a function because the input x = 3 produces two outputs; 4 and −1.
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If every vertical line intersects a graph at no more than one point, then the graph represents a function.
Vertical Line Test
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Example
Determine whether the graphs shown represent functions.a. b.
Passes the vertical line test.
The graph is a function.
Does not pass the vertical line test.
The graph is NOT a function.
