Copyright © 2010 Pearson Education, Inc. All rights reserved Sec

Copyright © 2010 Pearson Education, Inc. All rights reserved Sec Graphs of Linear Equations and Inequalities; Functions Chapter 3

Copyright © 2010 Pearson Education, Inc. All rights reserved Sec Introduction to Functions

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions Objectives 1.Define and identify relations and functions. 2.Find domain and range. 3.Identify functions defined by graphs and equations. 4.Use function notation. 5.Graph linear and constant functions.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions Terminology We often describe one quantity in terms of another. We can indicate the relationship between these quantities by writing ordered pairs in which the first number is used to arrive at the second number. Here are some examples. (5, $11) 5 gallons of gasolinewill cost $11. The total cost depends on the number of gallons purchased. (8, $17.60) 8 gallons of gasolinewill cost $ Again, the total cost depends on the number of gallons purchased.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions Terminology We often describe one quantity in terms of another. We can indicate the relationship between these quantities by writing ordered pairs in which the first number is used to arrive at the second number. Here are some examples. (the number of gallons, the total cost) depends on

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions Terminology We often describe one quantity in terms of another. We can indicate the relationship between these quantities by writing ordered pairs in which the first number is used to arrive at the second number. Here are some examples. (10, $150) Working for 10 hours,you will earn $150. The total gross pay depends on the number of hours worked. (15, $225) Working for 15 hours,you will earn $225. The total gross pay depends on the number of hours worked.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions Terminology We often describe one quantity in terms of another. We can indicate the relationship between these quantities by writing ordered pairs in which the first number is used to arrive at the second number. Here are some examples. (the number of hours worked, the total gross pay) depends on

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions Terminology We often describe one quantity in terms of another. We can indicate the relationship between these quantities by writing ordered pairs in which the first number is used to arrive at the second number. Here are some examples. ( x, y ) Generalizing, if the value of the variable y depends on the value of the variable x, then y is called the dependent variable and x is the independent variable. Independent variable Dependent variable

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions Define and identify relations and functions. Relation A relation is any set of ordered pairs. Function A function is a relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component. A special kind of relation, called a function, is very important in mathemat- ics and its applications.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 1Determining Whether Relations Are Functions Tell whether each relation defines a function. L = { (2, 3), (–5, 8), (4, 10) } M = { (–3, 0), (–1, 4), (1, 7), (3, 7) } N = { (6, 2), (–4, 4), (6, 5) } Relations L and M are functions, because for each different x -value there is exactly one y -value. In relation N, the first and third ordered pairs have the same x -value paired with two different y -values (6 is paired with both 2 and 5), so N is a relation but not a function. In a function, no two ordered pairs can have the same first component and different second components.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions Mapping Relations 1 F is a function. – F –1 G is not a function. –2 6 0 G

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions Tables and Graphs Graph of the function, F x y Table of the function, F x y – –6 O

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions Using an Equation to Define a Relation or Function y = –3 x Relations and functions can also be described using rules. Usually, the rule is given as an equation. For example, from the previous slide, the chart and graph could be described using the following equation. Independent variableDependent variable An equation is the most efficient way to define a relation or function.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec –12 15 –6 3.6 Introduction to Functions Functions NOTE Another way to think of a function relationship is to think of the indepen- dent variable as an input and the dependent variable as an output. This is illustrated by the input-output (function) machine (below) for the function defined by y = –3 x. y = –3 x (Input x ) (Output y ) (Input x ) –62 15–5 –124

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions 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.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 2Finding Domains and Ranges of Relations Give the domain and range of each relation. Tell whether the relation defines a function. (a) { (3, –8), (5, 9), (5, 11), (8, 15) } The domain, the set of x -values, is {3, 5, 8}; the range, the set of y -values, is {–8, 9, 11, 15}. This relation is not a function because the same x -value 5 is paired with two different y -values, 9 and 11.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 2Finding Domains and Ranges of Relations Give the domain and range of each relation. Tell whether the relation defines a function. (b) The domain of this relation is {6, 1, –9}. The range is {M, N}. This mapping defines a function – each x -value corresponds to exactly one y -value. 6 1 –9 M N

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 2Finding Domains and Ranges of Relations Give the domain and range of each relation. Tell whether the relation defines a function. (c) This is a table of ordered pairs, so the domain is the set of x -values, {–2, 1, 2}, and the range is the set of y -values, {3}. x y – The table defines a function because each different x -value corresponds to exactly one y -value (even though it is the same y -value).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 3Finding Domains and Ranges from Graphs Give the domain and range of each relation. (a) The domain is the set of x -values, {–3, 0, 2, 4}. The range, the set of y -values, is {–3, –1, 1, 2}. x y (–3, 2) (0, –3) (2, 1) (4, –1) O

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 3Finding 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 –7 and 2, inclusive. x y Domain Range The y -values include all numbers between –2 and 2, inclusive. Using interval notation, the domain is [–7, 2]; the range is [–2, 2]. O

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 3Finding 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 range include all real numbers, written (- ∞, ∞ ). x y O

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 3Finding Domains and Ranges from Graphs Give the domain and range of each relation. (d) The arrowheads indicate that the graph extends indefinitely left and right, as well as upward. The domain is (- ∞, ∞ ). x y Because there is a least y - value, –1, the range includes all numbers greater than or equal to –1, written [–1, ∞ ). O

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions Agreement on Domain The domain of a relation is assumed to be all real numbers that produce real numbers when substituted for the independent variable.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions Vertical Line Test If every vertical line intersects the graph of a relation in no more than one point, then the relation represents a function. y x y x Not a function – the same x -value corresponds to two different y -values. Function – each x -value corresponds to only one y -value. (a)(b)

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 4Using the Vertical Line Test Use the vertical line test to determine whether each relation is a function. (a) This relation is a function. x y (–3, 2) (0, –3) (2, 1) (4, –1) O

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 4Using the Vertical Line Test Use the vertical line test to determine whether each relation is a function. (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. x y O

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 4Using the Vertical Line Test Use the vertical line test to determine whether each relation is a function. (c) This relation is a function. x y O

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 4Using the Vertical Line Test Use the vertical line test to determine whether each relation is a function. (d) This relation is a function. x y O

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions 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.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 5Identifying Functions from Their Equations Decide whether each relation defines a function and give the domain. (a) y = x – 5 In the defining equation, y = x – 5, y is always found by subtracting 5 from x. Thus, 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 (– ∞, ∞ ).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 5Identifying Functions from Their Equations Decide whether each relation defines a function and give the domain. 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 defines a function. Since the equation involves a square root, the quantity under the radical sign cannot be negative. (b) y = 3 x – 1 3 x – 1 ≥ 0 3 x ≥ 1 x ≥, 1 3 and the domain of the function is [, ∞ ). 1 3 Thus,

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 5Identifying Functions from Their Equations Decide whether each relation defines a function and give the domain. The ordered pair (9, 3) and (9, –3) both satisfy this equation. Since one value of x, 9, corresponds to two values of y, 3 and –3, this equation does not define a function. Because x is equal to the square of y, the values of x (c) y = x 2 must always be nonnegative. The domain of the relation is [0, ∞ ).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 5Identifying Functions from Their Equations Decide whether each relation defines a function and give the domain. By definition, y is a function of x if every value of x leads to exactly one value of y. Here a particular value of x, say 4, corresponds to many values (d) y ≥ x – 3 of y. The ordered pairs (4, 7), (4, 6), (4, 5), and so on, all satisfy the inequality. Thus, an inequality never defines a function. be used for x so the domain is the set of real numbers (– ∞, ∞ ). Any number can

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 5Identifying Functions from Their Equations Decide whether each relation defines a function and give the domain. Given any value of x in the domain, we find y by adding 4, then dividing the result into 3. This process produces exactly one value of y for each value in the domain, so this equation defines a function. real numbers except those that make the denominator 0. We find these numbers by setting the denominator equal to 0 and solving for x. The domain includes all real numbers except –4, written (– ∞, –4) U (–4, ∞ ). The domain includes all (e) y = 3 x + 4 x + 4 = 0 x = –4

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions Variations of the Definition of Function 1. A function is a relation in which, for each 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 domain value.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions Function Notation When a function f 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 function notation, to express this and read f ( x ), as “f of x”. The letter f stands for function. For example, if y = 5 x – 2, we can name this function f and write f ( x ) = 5 x – 2. Note that f ( x ) is just another name for the dependent variable y.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec CAUTION The symbol f ( x ) does not indicate “f times x,” but represents the y -value for the indicated x -value. As shown below, f (3) is the y -value that corresponds to the x -value Introduction to Functions Function Notation y = f ( x ) = 5 x – 2 y = f (3) = 5(3) – 2 = 13

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 6Using Function Notation (a) f (4) Let f ( x ) = x + 2 x – 1. Find the following. 2 f ( x ) = x + 2 x – 1 2 f (4) = – 1 2 f (4) = – 1 f (4) = 23 Since f (4) = 23, the ordered pair (4, 23) belongs to f. Replace x with 4.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 6Using Function Notation (b) f ( w ) f ( x ) = x + 2 x – 1 2 f ( w ) = w + 2 w – 1 2 The replacement of one variable with another is important in later courses. Replace x with w. Let f ( x ) = x + 2 x – 1. Find the following. 2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 7Using Function Notation Let g ( x ) = 5 x + 6. Find and simplify g ( n + 2). g ( x ) = 5 x + 6 g ( n + 2) = 5( n + 2) + 6 = 5 n = 5 n + 16 Replace x with n + 2.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 8Using Function Notation For each function, find f (7). f ( x ) = – x + 2 f (7) = –7 + 2 = –5 Replace x with 7. (a) f ( x ) = – x + 2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 8Using Function Notation For each function, find f (7). (b) f = {(–5, –9), (–1, –1), (3, 7), (7, 15), (11, 23)} We want f (7), the y -value of the ordered pair where x = 7. As indicated by the ordered pair (7, 15), when x = 7, y = 15, so f (7) = 15.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 8Using Function Notation For each function, find f (7). (c) The domain element 7 is paired with 17 in the range, so f (7) = f DomainRange 717

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec x y O 3.6 Introduction to Functions EXAMPLE 8Using Function Notation For each function, find f (7). (d) To evaluate f (7), find 7 on the x -axis Then move up until the graph of f is reached. tally to the y -axis gives 3 for the corresponding y -value. Thus, f (7) = 3. Moving horizon

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions Finding an Expression for f ( x ) Step 1 Solve the equation for y. Step 2 Replace y with f ( x ).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 9Writing Equations Using Function Notation Rewrite each equation using function notation. Then find f (–3) and f ( n ). This equation is already solved for y. Since y = f ( x ), (a) y = x – 1 2 f ( x ) = x – 1. 2 To find f (–3), let x = –3. f (–3) = (–3) – 1 2 = 9 – 1 = 8 To find f ( n ), let x = n. f ( n ) = n – 1 2

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions EXAMPLE 9Writing Equations Using Function Notation Rewrite each equation using function notation. Then find f (–3) and f ( n ). First solve x – 5 y = 3 for y. Then replace y with f ( x ). (b) x – 5 y = 3 x – 5 y = 3 x – 3 = 5 y y = x – 3 5 so Add 5 y ; subtract 3. Now find f (–3) and f ( n ). f (–3) – (–3) = 6 5 – = f ( n ) – (n)(n) = Let x = –3 Let x = n f ( x ) – x =

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions Linear Function A function that can be defined by f ( x ) = ax + b, for real numbers a and b is a linear function. The value of a is the slope of m of the graph of the function.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Introduction to Functions Linear Function A linear function defined by f ( x ) = b (whose graph is a horizontal line) is sometimes called a constant function. The domain of any linear function is (– ∞, ∞ ). The range of a nonconstant linear function is (– ∞, ∞ ), while the range of the constant function defined by f ( x ) = b is { b }.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec O 3.6 Introduction to Functions Linear Function Recall that m is the slope of the line and (0, b ) is the y -intercept. In Example 9 (b), we wrote x – 5 y = 3 as the linear function defined by f ( x ) – x = Slope x y To graph this function, plot the y -intercept and use the definition of slope as to find a second point on the line. rise run Draw a straight line through these points to obtain the graph. 3 5 – y -intercept is (0, ). 3 5 – m = 1 5