M098
Carson Elementary and Intermediate Algebra 3e
Section 4.7
Objectives
1.
2.
Identify the domain and range of a relation.
Identify functions and their domains and ranges.
3.
4.
Find the value of a function.
Graph linear functions.
Vocabulary
Relation
Domain
Range
Function
Function notation
A set of ordered pairs.
The set of all input values (x-values) for a relation.
The set of all output values (y-values) for a relation.
A relation in which every value in the domain is paired with exactly one value in the
range.
f(x), read as f of x.
Prior Knowledge
Graphing lines using slope intercept form.
New Concepts
1. Identify the domain and range of a relation.
A relation is any set of ordered pairs.
The domain is the set of input or x values.
The range is set of all output or y values.
Example 1:
Determine the domain and range of the relation.
{(-2,1), (4,3), (2,5), (-3,-2)}
Example 2:
Domain: {-2, 4, 2, -3}
Range: {1, 3, 5, -2}
Determine the domain and range of the relation.
{(5,4), (-3, 8), (5, -2), (0, -5)}
Domain: {5, -3, 0}
Range: {4, 8, -2, -5}
The domain and range can also be determined from a graph.
Domain: move from left to right on the x-axis.
Range: move from bottom to top on the y-axis.
Example 3:
Determine the domain and range of the relation.
Domain: {x | x ≤ 3 } or (-∞, 3]
Range: {y | y ≥ -1 } or [-1, ∞)
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M098
Carson Elementary and Intermediate Algebra 3e
Example 4:
Section 4.7
Determine the domain and range of the relation.
Domain: {x | x is a real number } or (-∞, ∞)
Range: {y | y ≤ 4 } or (-∞, 4]
2. Identify functions and their domains and ranges.
A function is a special relation in which each input value (x-value) is paired with only one output value
(y-value). In other words, the x-coordinates can not repeat. It is okay for the y-coordinates to repeat.
Example 5:
Example 1:
Example 2:
Example 6:
Look back at Examples 1 – 2. Which of those relations are functions?
Function
Not a Function
Determine if the relation is a function and give the domain and range.
{(-1,5), (2,6), (2,9), (-3,-2)}
Domain: {-1, 2, -3}
Range: {5, 6, 9, -2}
Function: No
Example 7:
Determine if the relation is a function and give the domain and range.
{(1,2), (4,-7), (9,5), (6,14)}
Domain: {1, 4, 9, 6}
Range: {2, -7, 5, 14}
Function: Yes
The vertical line test helps to determine if a relation is a function from its graph. Imagine a vertical line
through every point in the domain (x-value) of the graph. If none of the lines intersect the graph at
more than one point, the relation is a function.
Example 8:
Example 3
Example 4:
Example 9:
Look back at Examples 3 – 4. Which of those relations are functions?
Function
Function
Determine if the relation is a function and give the domain and range.
Domain: {x | -1 ≤ x ≤ 4} or [-1, 4]
Range: { y | -3 ≤ y ≤ 2 } or [-3, 2]
Function: No
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M098
Carson Elementary and Intermediate Algebra 3e
Example 10:
Section 4.7
Determine if the relation is a function and give the domain and range.
Domain: {x | any real number} or (-∞, ∞)
Range: { y | any real number} or (-∞, ∞)
Function: Yes
Example 11: Determine if the relation is a function. The domain is in the left column
and the range is in the right column.
Average rainfall for Seattle:
Avg. Rain (inches)
6
4
3
2
1
Month
January, December
February, March
October
April, May, September
June, July, August
Function: No
3. Find the value of a function.
Function notation f(x) – read f of x – replaces the variable y. This notation says that
(1) the relation is a function and
(2) the variable is x.
y and f(x) are just two ways of naming the same relation. We use whichever is more convenient for
what we are doing. Students are used to seeing (x, y) notation and are comfortable with it. Now they
need to become comfortable with (x, f(x) ). Remember that y and f(x) refer to the same thing.
Much of the work that is done with functions involves evaluating the function for different input (x)
values. Function notation (f(x)) provides a convenient shorthand notation for “Evaluate this function
when x equals ___.”
Other letters besides f can be used in function notation. Other commonly used letters are g(x) and h(x).
Example 12:
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f(x) = 5x + 3
a.
f(1) = 5(1) + 3
f(1) = 5 + 3
f(1) = 8
b.
f(0) = 5(0) + 3
f(0) = 0 + 3
f(0) = 3
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M098
Carson Elementary and Intermediate Algebra 3e
Section 4.7
2
f(x) = 3x + 2x – 1
Example 13:
2
a.
f(-2) = 3(-2) + 2(-2) – 1
f(-2) = 3(4) – 4 – 1
f(-2) = 7
b.
f(.2) = 3(.2) + 2(.2) – 1
f(.2) = -0.48
c.
f(a) = 3a + 2a – 1
2
2
f x  
Example 14:
f  2 
a.
x5
25
f  2   7
Not a real number
b.
f 6  
65
f 6  
1
f 6   1
f a 
c.
Example 15:
a5
f x  
2
x2
2
22
2
f  2 
0
Undefined
f  2 
a.
2
32
2
f 3  
5
f 3  
b.
A graph can also be used to determine the value of the function.
Example 16:
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a.
f(0) = 5
b.
f(2) = 0
c.
f(-2) = undefined
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M098
Carson Elementary and Intermediate Algebra 3e
Section 4.7
4. Graph linear functions.
Remember that f(x) and y are interchangeable. To graph a linear function, rewrite the function using y
instead of f(x). That puts the equation into slope-intercept form, which gives the slope and the yintercept.
Example 17:
Graph f(x) = -x – 1
Replace f(x) with y.
y = -x – 1
This is slope intercept form.
m = -1
y-intercept = (0, -1)
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