TH EDITION LIAL HORNSBY SCHNEIDER COLLEGE ALGEBRA

Functions Relations and Functions Domain and Range Determining Functions from Graphs or Equations Function Notation Increasing, Decreasing, and Constant Functions

Relation A relation is a set of ordered pairs.

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.

Motion Problems Note The relation from the beginning of this section representing the number of gallons of gasoline and the corresponding cost is a function since each x-value is paired with exactly one y-value. You would not be happy if you and a friend each pumped 10 gal of regular gasoline at the same station and your bills were different.

Example 1 DECIDING WHETHER RELATIONS DEFINE FUNCTIONS Decide whether the relation defines a function. 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. x-values of F y-values of F

Example 1 DECIDING WHETHER RELATIONS DEFINE FUNCTIONS Decide whether the relation defines a function. Solution As the correspondence shows below, relation G is not a function because one first component corresponds to more than one second component. x-values of G y-values of G

Example 1 DECIDING WHETHER RELATIONS DEFINE FUNCTIONS Decide whether the relation defines a function. 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. Same x-values Different y-values Not a function

Mapping Relations and functions can also be expressed as a correspondence or mapping from one set to another. In the example below the arrows from 1 to 2 indicates that the ordered pair (1, 2) belongs to F. Each first component is paired with exactly one second component. 1– 231– 23 24– 124– 1 x-axis valuesy-axis values

Mapping In the mapping for relations H, which is not a function, the first component – 2 is paired with two different second components, 1 and 0. – 4– 2– 4– x-axis valuesy-axis values

Note Another way to think of a function relationship is to think of the independent variable as an input and the dependent variable as an output. Relations

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.

Example 2 FINDING DOMAINS AND RANGES OF RELATIONS Give the domain and range of the relation. Tell whether the relation defines a function. a. 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.

Example 2 FINDING DOMAINS AND RANGES OF RELATIONS Give the domain and range of the relation. Tell whether the relation defines a function. b. 467– 3467– 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.

Example 2 FINDING DOMAINS AND RANGES OF RELATIONS Give the domain and range of the relation. Tell whether the relation defines a function. c. This relation is a set of ordered pairs, so the domain is the set of x- values {– 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. xy – 5–

Example 3 FINDING DOMAINS AND RANGES FROM GRAPHS Give the domain and range of each relation. a. (– 1, 1) (1, 2) (0, – 1) (4, – 3) 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}. x y

Example 3 FINDING DOMAINS AND RANGES FROM GRAPHS Give the domain and range of each relation. b. The x-values of the points on the graph include all numbers between – 4 and 4, inclusive. The y- values include all numbers between – 6 and 6, inclusive. 4 – 4– 4 6 – 6– 6 x y The domain is [– 4, 4]. The range is [– 6, 6].

Example 3 FINDING DOMAINS AND RANGES FROM GRAPHS Give the domain and range of each relation. c. 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 (– ,  ). x y

Example 3 FINDING DOMAINS AND RANGES FROM GRAPHS Give the domain and range of each relation. d. The arrowheads indicate that the line extends indefinitely left and right, as well as upward. The domain is (– ,  ). Because there is at least y-value, – 3, the range includes all numbers greater than, or equal to – 3 or [– 3,  ). x y

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.

Vertical Line Test If each vertical line intersects a graph in at most one point, then the graph is that of a function.

Example 4 USING THE VERTICAL LINE TEST a. (– 1, 1) (1, 2) (0, – 1) (4, – 3) This graph represents a function. Use the vertical line test to determine whether each relation graphed is a function. x y

Example 4 b. 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. 4 – 4– 4 6 – 6– 6 x y USING THE VERTICAL LINE TEST Use the vertical line test to determine whether each relation graphed is a function.

Example 4 c. This graph represents a function. x y USING THE VERTICAL LINE TEST Use the vertical line test to determine whether each relation graphed is a function.

Example 4 d. This graph represents a function. x y USING THE VERTICAL LINE TEST Use the vertical line test to determine whether each relation graphed is a function.

Relations Note Graphs that do not represent functions are still relations. Remember that all equations and graphs represent relations and that all relations have a domain and range.

Example 5 IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Decide whether each relation defines a function and give the domain and range. Solution Since y is always found by adding 4 to x, each value of x corresponds to just one value of y and the relation defines a function. x can be any real number, so the domain is a. or Since y is always 4 more than x, y may also be any real number, and so the range is

Example 5 IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Decide whether each relation defines a function and give the domain and range. 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. b.

Example 5 IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Decide whether each relation defines a function and give the domain and range. Solution b. Solve the inequality. Add 1. Divide by 2. Domain is

Example 5 IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Decide whether each relation defines a function and give the domain and range. Solution b. 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

Example 5 IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Decide whether each relation defines a function and give the domain and range. Solution Ordered pairs (16, 4) and (16, – 4) both satisfy the equation. Since one value of x, 16, corresponds to two values of y, this equation does not define a function. c. The domain is Any real number can be squared, so the range of the relation is

Example 5 IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Decide whether each relation defines a function and give the domain and range. Solution The ordered pairs (1, 0), (1, – 1), (1, – 2), and (1, – 3) all satisfy the inequality. An inequality rarely defines a function. Since any number can be used for x or for y, the domain and range are the set of real numbers or d.

Example 5 IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Decide whether each relation defines a function and give the domain and range. 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. e.

Example 5 IDENTIFYING FUNCTIONS, DOMAINS, AND RANGES Decide whether each relation defines a function and give the domain and range. Solution Domain includes all real numbers except those making the denominator 0. Add 1. The domain includes all real numbers except 1 and is written The range is the interval e.

Function Notation When a function  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. called a function notation, to express this and read  (x) as “  of x.” The letter  is he name given to this function. For example, if y = 9x – 5, we can name the function  and write

Function Notation Note that  (x) is just another name for the dependent variable y. Fore example, if y =  (x) = 9x – 5 and x = 2, then we find y, or  (2), by replacing x with 2. The statement “if x = 2, the y = 13” represents the ordered pair (2, 13) and is abbreviated with the function notation as

Function Notation Read “  of 2” or “  at 2.” Also, and These ideas can be illustrated as follows. Value of the function Name of the functionDefining expression Name of the independent variable

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.

Caution The symbol  (x) does not indicate “  times x,” but represents the y- value for the indicated x-value. As just shown,  (2) is the y-value that corresponds to the x-value 2.

Example 6 USING FUNCTION NOTATION Let  (x) = – x 2 + 5x – 3 and g (x) = 2x + 3. Find and simplify. Solution a. Replace x with 2. Apply the exponent; multiply. Add and subtract. Thus,  (2) = 3; the ordered pair (2, 3) belongs to .

Example 6 USING FUNCTION NOTATION Let  (x) = – x 2 + 5x – 3 and g (x) = 2x + 3. Find and simplify. Solution b. Replace x with q.

Example 6 USING FUNCTION NOTATION Let  (x) = x 2 + 5x – 3 and g (x) = 2x + 3. Find and simplify. Solution c. Replace x with a + 1.

Example 7 USING FUNCTION NOTATION For each function, find  (3). Solution a. Replace x with 3.

Example 7 USING FUNCTION NOTATION For each function, find  (3). Solution For  = {( – 3, 5), (0, 3), (3, 1), (6, – 9)}, we want  (3), the y-value of the ordered pair where x = 3. As indicated by the ordered pair (3, 1), when x = 3, y = 1,so  (3) = 1. b.

Example 7 USING FUNCTION NOTATION For each function, find  (3). Solution c. – DomainRange In the mapping, the domain element 3 is paired with 5 in the range, so  (3) = 5.

Example 7 USING FUNCION NOTATION For each function, find  (3). Solution d. Start at 3 on the x-axis and move up to the graph. Then, moving horizontally to the y- axis gives 4 for the corresponding y-value. Thus  (3) =

Finding an Expression for  (x) Consider an equation involving x and y. Assume that y can be expressed as a function  of x. To find an expression for  (x): 1.Solve the equation for y. 2.Replace y with  (x).

Example 8 WRITING EQUATIONS USING FUNCTION NOTATION Solution a. Assume that y is a function of x. Rewrite the function using notation. Let y =  (x) Now find  (– 2) and  (a). Let x = – 2

Example 8 WRITING EQUATIONS USING FUNCTION NOTATION Solution a. Assume that y is a function of x. Rewrite the function using notation. Let y =  (x) Now find  (– 2) and  (a). Let x = a

Example 8 WRITING EQUATIONS USING FUNCTION NOTATION Solution b. Assume that y is a function of x. Rewrite the function using notation. Solve for y. Multiply by – 1; divide by 4.

Example 8 WRITING EQUATIONS USING FUNCTION NOTATION Solution b. Assume that y is a function of x. Rewrite the function using notation. Now find  (– 2) and  (a). Let x = – 2 Let x = a

Increasing, Decreasing, and Constant Functions Suppose that a function  is defined over an interval I. If x 1 and x 2 are in I, (a)  increases on I if, whenever x 1 < x 2,  (x 1 ) <  (x 2 ) (b)  decreases on I if, whenever x 1  (x 2 ) (c)  is constant on I if, for every x 1 and x 2,  (x 1 ) =  (x 2 )

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 – 2– 2 x y

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 – 2– 2 On the interval (– , 1), the y-values are decreasing; on the interval [1,3], the y- values are increasing; on the interval [3,  ), the y- values are constant (and equal to 6). Solution x y

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 – 2– 2 Therefore, the function is decreasing on (– , 1), increasing on [1,3], and constant on [3,  ). Solution x y

Example 10 INTERPRETING A GRAPH Hours Gallons Swimming Pool Water Level This graph shows the relationship between the number of gallons, g (t), of water in a small swimming pool and time in hours, t. Answer the following questions using the graph information.

Example 10 INTERPRETING A GRAPH Solution The max range value is 3000 and the max number of gallons, 3000, is first reached at t = 25 hr. a. What is the maximum number of gallons of water in the pool? When is the maximum water level first reached? Hours Gallons Swimming Pool Water Level

Example 10 INTERPRETING A GRAPH Solution The water level is increasing for 25 – 0 = 25 hr and is decreasing for 75 – 50 = 25 hr. It is constant for (50 – 25) + (100 – 75) = = 50 hr. b. For how long is the water level increasing? Decreasing? Constant? Hours Gallons Swimming Pool Water Level

Example 10 INTERPRETING A GRAPH Solution When t = 90, y = g (90) = There are 2000 gal after 90 hr. c. How many gallons of water are in the pool after 90 hr? Hours Gallons Swimming Pool Water Level

Example 10 INTERPRETING A GRAPH Solution The pool is empty at the beginning, then filled to a level of 3000 gal during the first 25 hr and the water level then remains the same. At 50 hr the pool starts to be drained over 25 hr to 2000 gal and remains there for 25 hr. d. Describe a series of events that could account for the water level changes shown in the graph Hours Gallons Swimming Pool Water Level