2.3 Functions Relations and Functions Domain and Range Determining Functions from Graphs or Equations Function Notation Increasing, Decreasing, and Constant Functions
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Relation A relation is a set of ordered pairs.
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Function A function is a relation in which, for each distinct value of the first component of the ordered pair, there is exactly one value of the second component.
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Example 1
DECIDING WHETHER RELATIONS DEFINE FUNCTIONS
Decide whether the relation defines a function. F = {(1,2),( −2,4)(3, −1)} Solution Relation F is a function, because for each different x-value there is exactly one y-value. We can show this correspondence as follows.
{1, − 2, 3}
x-values of F
{2, 4, −1}
y-values of F 2.3 - 4
Example 1
DECIDING WHETHER RELATIONS DEFINE FUNCTIONS
Decide whether the relation defines a function. G = {(1,1),(1,2)(1,3)(2,3)} Solution As the correspondence shows below, relation G is not a function because one first component corresponds to more than one second component.
{1, 2}
x-values of G
{1, 2, 3}
y-values of G 2.3 - 5
Example 1
DECIDING WHETHER RELATIONS DEFINE FUNCTIONS
Decide whether the relation defines a function. H =− {( 4,1),( −2,1)( −2,0)} Solution In relation H the last two ordered pairs have the same x-value paired with two different y-values, so H is a relation but not a function. Different y-values
H=
{( − 4,1),( −2,1)( −2,0)}
Not a function
Same x-values 2.3 - 6
Domain and Range In a relation, the set of all values of the independent variable (x) is the domain. The set of all values of the dependent variable (y) is the range.
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Example 2
FINDING DOMAINS AND RANGES OF RELATIONS
Give the domain and range of the relation. Tell whether the relation defines a function. a. {(3, −1),(4,2),(4,5),(6,8)} The domain, the set of x-values, is {3, 4, 6}; the range, the set of y-values is {–1, 2, 5, 8}. This relation is not a function because the same x-value, 4, is paired with two different y-values, 2 and 5. 2.3 - 8
Example 2
FINDING DOMAINS AND RANGES OF RELATIONS
Give the domain and range of the relation. Tell whether the relation defines a function. b.
4 6 7 –3
100 200 300
The domain is {4, 6, 7, –3}; the range is {100, 200, 300}. This mapping defines a function. Each x-value corresponds to exactly one y-value. 2.3 - 9
Example 2
FINDING DOMAINS AND RANGES OF RELATIONS
Give the domain and range of the relation. Tell whether the relation defines a function. c.
x –5 0 5
y 2 2 2
This relation is a set of ordered pairs, so the domain is the set of xvalues {–5, 0, 5} and the range is the set of y-values {2}. The table defines a function because each different x-value corresponds to exactly one y-value. 2.3 - 10
Example 3
FINDING DOMAINS AND RANGES FROM GRAPHS
Give the domain and range of each relation. y
a. (1, 2) (– 1, 1) x (0, – 1)
The domain is the set of x-values which are {– 1, 0, 1, 4}. The range is the set of y-values which are {– 3, – 1, 1, 2}.
(4, – 3)
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FINDING DOMAINS AND RANGES FROM GRAPHS
Example 3
Give the domain and range of each relation. y
b.
6
x
–4
4
The x-values of the points on the graph include all numbers between –4 and 4, inclusive. The yvalues include all numbers between –6 and 6, inclusive. The domain is [–4, 4]. The range is [–6, 6].
–6 2.3 - 12
FINDING DOMAINS AND RANGES FROM GRAPHS
Example 3
Give the domain and range of each relation. y
c.
x
The arrowheads indicate that the line extends indefinitely left and right, as well as up and down. Therefore, both the domain and the range include all real numbers, written (– ∞, ∞). 2.3 - 13
FINDING DOMAINS AND RANGES FROM GRAPHS
Example 3
Give the domain and range of each relation. y
d.
The arrowheads indicate that the line extends indefinitely left and right, as well as upward. The domain is (– ∞, ∞). Because there is at least x y-value, –3, the range includes all numbers greater than, or equal to –3 or [–3, ∞).
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Agreement on Domain Unless specified otherwise, the domain of a relation is assumed to be all real numbers that produce real numbers when substituted for the independent variable.
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Vertical Line Test If each vertical line intersects a graph in at most one point, then the graph is that of a function.
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Example 4
USING THE VERTICAL LINE TEST
Use the vertical line test to determine whether each relation graphed is a function. y
a. (1, 2) (– 1, 1)
This graph represents a x function.
(0, – 1) (4, – 3)
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USING THE VERTICAL LINE TEST
Example 4
Use the vertical line test to determine whether each relation graphed is a function. y
b.
6
x
–4
4
This graph fails the vertical line test, since the same x-value corresponds to two different y-values; therefore, it is not the graph of a function.
–6 2.3 - 18
USING THE VERTICAL LINE TEST
Example 4
Use the vertical line test to determine whether each relation graphed is a function. y
c.
x
This graph represents a function.
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USING THE VERTICAL LINE TEST
Example 4
Use the vertical line test to determine whether each relation graphed is a function. y
d.
x
This graph represents a function.
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IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Decide whether each relation defines a function and give the domain and range. Example 5
y b.=
2x − 1
Solution For any choice of x in the domain, there is exactly one corresponding value for y (the radical is a nonnegative number), so this equation is a function. Since the equation involves a square root, the quantity under the radical cannot be negative.
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IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Decide whether each relation defines a function and give the domain and range. Example 5
y b.=
2x − 1
Solution
2x − 1 ≥ 0 2x ≥ 1 1 x≥ 2
Solve the inequality. Add 1. Divide by 2.
1 Domain is , ∞ . 2 2.3 - 22
IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Decide whether each relation defines a function and give the domain and range. Example 5
y b.=
2x − 1
Solution
2x − 1 ≥ 0 2x ≥ 1 1 x≥ 2
Solve the inequality. Add 1. Divide by 2.
Because the radical is a non-negative number, as x takes values greater than or equal to ½ , the range is y ≥ 0 or [0, ∞ ) .
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IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Decide whether each relation defines a function and give the domain and range. Example 5
5 e. y = x −1 Solution Substituting any value in for x, subtracting 1 and then dividing it into 5, produces exactly one value of y for each value in the domain. This equation defines a function.
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IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Decide whether each relation defines a function and give the domain and range. Example 5
5 e. y = x −1 Solution Domain includes all real numbers except those making the denominator 0.
x −1= 0 x =1
Add 1.
The domain includes all real numbers except 1 and is written ( − ∞,1) ∪ (1, ∞ ) . The range is the interval ( − ∞,0 ) ∪ ( 0 , ∞ ) . 2.3 - 25
Function Notation When a function ƒ is defined with a rule or an equation using x and y for the independent and dependent variables, we say “y is a function of x” to emphasize that y depends on x. We use the notation.
y = f ( x ),
called a function notation, to express this and read ƒ(x) as “ƒ of x.” The letter ƒis he name given to this function. For example, if y = 9x – 5, we can name the function ƒ and write
) 9 x − 5. f ( x= 2.3 - 26
Function Notation Note that ƒ(x) is just another name for the dependent variable y. Fore example, if y = ƒ(x) = 9x – 5 and x = 2, then we find y, or ƒ(2), by replacing x with 2.
f (2)= 9 2 − 5= 13 The statement “if x = 2, the y = 13” represents the ordered pair (2, 13) and is abbreviated with the function notation as
f (2) = 13. 2.3 - 27
Function Notation f (2) = 13 Read “ƒ of 2” or “ƒ at 2.” Also,
3) 9( −3) −= 5 −32. f (0) = 9 0 − 5 = −5 and f ( −= These ideas can be illustrated as follows. Name of the function
= y Value of the function
Defining expression
= f (x)
9x − 5
Name of the independent variable 2.3 - 28
Variations of the Definition of Function 1. A function is a relation in which, for each distinct value of the first component of the ordered pairs, there is exactly one value of the second component. 2. A function is a set of ordered pairs in which no first component is repeated. 3. A function is a rule or correspondence that assigns exactly one range value to each distinct domain value. 2.3 - 29
Caution The symbol ƒ(x) does not indicate “ƒ times x,” but represents the yvalue for the indicated x-value. As just shown, ƒ(2) is the y-value that corresponds to the x-value 2.
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Example 6
USING FUNCTION NOTATION
Let ƒ(x) = –x2 + 5x – 3 and g(x) = 2x + 3. Find and simplify. a. ƒ(2) Solution
ƒ( x ) = −x 2 + 5x − 3 ƒ(2) = −22 + 5 2 − 3 =− 4 + 10 − 3 =3
Replace x with 2. Apply the exponent; multiply. Add and subtract.
Thus, ƒ(2) = 3; the ordered pair (2, 3) belongs to ƒ. 2.3 - 31
Example 6
USING FUNCTION NOTATION
Let ƒ(x) = –x2 + 5x – 3 and g(x) = 2x + 3. Find and simplify. b. ƒ(q ) Solution
ƒ( x ) = −x 2 + 5x − 3 ƒ (q ) = −q 2 + 5q − 3
Replace x with q.
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Example 6
USING FUNCTION NOTATION
Let ƒ(x) = x2 + 5x –3 and g(x) = 2x + 3. Find and simplify. c. g (a + 1) Solution
g ( x= ) 2x + 3 (a + 1) 2(a + 1) + 3 g= = 2a + 2 + 3
Replace x with a + 1.
= 2a + 5 2.3 - 33
Example 7
USING FUNCION NOTATION
For each function, find ƒ(3). y = ƒ( x )
d. Solution Start at 3 on the x-axis and move up to the graph. Then, moving horizontally to the yaxis gives 4 for the corresponding y-value. Thus ƒ(3) = 4.
4 2 0
2
3 4
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Example 8
WRITING EQUATIONS USING FUNCTION NOTATION
Assume that y is a function of x. Rewrite the function using notation. a. = y x2 + 1 2 y x +1 = Solution ƒ( x= ) x2 + 1
Let y = ƒ(x)
Now find ƒ(–2) and ƒ(a). ƒ( −2) = ( −2) + 1 2
Let x = –2
= 4 +1 =5 2.3 - 35
Example 8
WRITING EQUATIONS USING FUNCTION NOTATION
Assume that y is a function of x. Rewrite the function using notation. a. = y x2 + 1 2 y x +1 = Solution ƒ( x= ) x2 + 1
Let y = ƒ(x)
Now find ƒ(–2) and ƒ(a). ƒ (a ) = a 2 + 1
Let x = a
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Example 8
WRITING EQUATIONS USING FUNCTION NOTATION
Assume that y is a function of x. Rewrite the function using notation. 5 b. x − 4 y = Solution
x − 4y = 5
Solve for y.
− 4 y =− x + 5 x −5 y= 4 1 5 ƒ( x ) = x − 4 4
Multiply by –1; divide by 4.
a−b a b = − c c c 2.3 - 37
Example 8
WRITING EQUATIONS USING FUNCTION NOTATION
Assume that y is a function of x. Rewrite the function using notation. 5 b. x − 4 y = Solution Now find ƒ(–2) and ƒ(a).
1 5 7 ƒ( −2) = ( −2) − = − 4 4 4 1 5 ƒ (a ) = a − 4 4
Let x = –2
Let x = a
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Increasing, Decreasing, and Constant Functions Suppose that a function ƒ is defined over an interval I. If x1 and x2 are in I, (a) ƒ increases on I if, whenever x1 < x2, ƒ(x1) < ƒ(x2) (b) ƒ decreases on I if, whenever x1 < x2, ƒ(x1) > ƒ(x2) (c) ƒ is constant on I if, for every x1 and x2, ƒ(x1) = ƒ(x2) 2.3 - 39
Example 9
DETERMINING INTERVALS OVER WHICH A FUNCTION IS INCREASING, DECREASING, OR CONSTANT
Determine the intervals over which the function is increasing, decreasing, or constant. y
6
2
x –2
1
3
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DETERMINING INTERVALS OVER WHICH A FUNCTION IS INCREASING, DECREASING, OR CONSTANT
Example 9
Determine the intervals over which the function is increasing, decreasing, or constant. y Solution 6
2 –2
1
3
On the interval (–∞, 1), the y-values are decreasing; on the interval [1,3], the yvalues are increasing; on x the interval [3, ∞), the yvalues are constant (and equal to 6).
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DETERMINING INTERVALS OVER WHICH A FUNCTION IS INCREASING, DECREASING, OR CONSTANT
Example 9
Determine the intervals over which the function is increasing, decreasing, or constant. y Solution 6
2 –2
1
3
Therefore, the function is decreasing on (–∞, 1), x increasing on [1,3], and constant on [3, ∞).
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