Transcript COL_4.1
HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems: College Algebra Section 4.1: Relations and Functions HAWKES LEARNING SYSTEMS Copyright © 2011 Hawkes Learning Systems. All rights reserved. math courseware specialists Objectives o o o o Relations, domains and ranges. Functions and the vertical line test. Functional notation and function evaluation. Implied domain of a function. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Relations, Domains and Ranges Relations, Domains and Ranges o A relation is a set of ordered pairs. Any set of ordered pairs automatically relates the set of first coordinates to the set of second coordinates, and these sets have special names. o The domain of a relation is the set of all the first coordinates. o The range of a relation is the set of all second coordinates. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 1: Relations, Domains and Ranges The set R 4, 2 , 6, 1 , 0, 0 , 4, 0 , , 2 is a relation consisting of five ordered pairs. What is the domain? R 4 , 2 , 6 , 1 , 0 , 0 , 4 , 0 , , 2 4, 6, 0, Note: it is not necessary to list – 4 twice in the domain. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 1: Relations, Domains and Ranges The set R 4, 2 , 6, 1 , 0, 0 , 4, 0 , , 2 is a relation consisting of five ordered pairs. 2 R 4, 2 , 6, 1 , 0, 0 , 4, 0 , , What is the range? 2, 1, 0, 2 Again, you do not have to write 0 twice in the range. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 2: Relations, Domains and Ranges The equation 3 x 7 y 13 describes a relation. This relation consists of an infinite number of ordered pairs, so it is not possible to list them all as a set. Ex: One of the ordered pairs of this relation is 2,1 , since 3 2 7 1 1 3 . Both the domain and range of this relation are the set of real numbers. The graph of this relation is shown to the right. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 3: Relations, Domains and Ranges The picture below describes a relation with infinite elements. What is the domain? 1,1 What is the range? 2, 2 HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 4: Relations, Domains and Ranges The picture below describes a similar relation to the one in example 3. The shading indicates that all ordered pairs inside the rectangle, as well as the ordered pairs laying on the rectangle, are elements of the relation. What is the domain? 1,1 What is the range? 2, 2 Note: The range and domain are the same as in example 3. Does this mean the relations are the same? HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 5: Relations, Domains and Ranges The set S x , y | x is the ow ner of y is a relation among domestic dogs. What is the domain? The set of all owners. What is the range? The set of all dogs that are pets. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Functions and the Vertical Line Test Functions A function is a relation in which every element of the domain is paired with exactly one element of the range. Equivalently, a function is a relation in which no two distinct ordered pairs have the same first coordinate. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Functions and the Vertical Line Test Ex: Which of the following relations is a function? HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Functions and the Vertical Line Test The Vertical Line Test If a relation can be graphed in the Cartesian plane, the relation is a function if and only if no vertical line passes through the graph more than once. If even one vertical line intersects the graph of the relation two or more times, the relation fails to be a function. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Functions and the Vertical Line Test Ex: Which coordinates of the relation R 7,1 , 3, 1 , 0, 0 , 3, 0 , 5, 2 fail the vertical line test and, therefore, do NOT define the relation as a function? 3, 1 (3, 0) HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 6: Functions and the Vertical Line Test Consider the relation: R 3, 2 , 1, 0 , 0, 2 , 2, 4 , 4, 0 Do any vertical lines intersect the graph twice or more at any one point? No Is the relation a function? Y es HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 7: Functions and the Vertical Line Test Do any vertical lines intersect the graph twice or more at any one point? Y es Is the relation below a function? No HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 8: Functions and the Vertical Line Test Do any vertical lines intersect the graph twice or more times at any one point? No Is the relation below a function? Y es HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Functional Notation and Function Evaluation Functional Notation Functional notation is a means of writing a function in terms of x. Ex. Given the function 2 x y 1, solve for y: y 2 x 1 . In this context, x is called the independent variable and y is the dependent variable. In functional notation, we would write the function above as f x 2 x 1 , read “f of x equals negative two x plus one.” f x 2 x 1 indicates that when given a specific value for x, the function f returns negative two times that value, plus 1. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Functional Notation and Function Evaluation Caution! A common error is to think that f(x) stands for the product of f and x. This is wrong! While it is true that parentheses often indicate multiplication, they are also used in defining functions. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Functional Notation and Function Evaluation In defining the function f as f ( x ) 2 x 1 , the critical idea being conveyed is the formula. We can use absolutely any symbol as a placeholder in defining f. For instance, f ( x ) 2 x 1, f ( p ) 2 p 1, f ( ) 2 1 all define the same function. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Functional Notation and Function Evaluation The variable, or symbol, that is used in defining a given function is called its argument and serves as nothing more than a placeholder. Ex: What is the argument for each function? f ( x ) 2 x 1 The argument is x. f ( p ) 2 p 1 The argument is p. f ( ) 2 1 The argument is . HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 9: Functional Notation and Function Evaluation The following equation in x and y represents a function. Rewrite the equation in functional notation, then evaluate for x = 5. 12 x 4 3 y 2 First, solve the equation for y. 12 x 3 y 6 3 y 12 x 6 y 4x 2 f x 4x 2 f 5 4 5 2 f (5) 22 We can name the function any number of names. Here, we choose f, the same name in the previous examples. We now evaluate f at 5 for the desired answer. HAWKES LEARNING SYSTEMS Copyright © 2011 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 10: Functional Notation and Function Evaluation The following equations in x and y represent a function. Rewrite the equations in functional notation, then evaluate for x = – 2. First, solve the a. y 7 x 2 equation for y. y x 7 2 g x x 7 2 g 2 2 7 2 g 2 4 7 Here we named the function g. We now evaluate g at – 2 for the desired answer. g 2 11 Continued on the next slide… HAWKES LEARNING SYSTEMS Copyright © 2011 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 10: Functional Notation and Function Evaluation (cont.) The following equations in x and y represent a function. Rewrite the equations in functional notation, then evaluate for x = – 2. Solve as usual. b. 79 x y 9 y 9 79 x y 9 79 x h x 9 79 x h 2 9 h 2 9 9 h 2 0 79 2 HAWKES LEARNING SYSTEMS Copyright © 2011 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 11: Functional Notation and Function Evaluation Given the function f x 2 x 2 3, evaluate: a. f b 2b 3 2 Simply replace x with b. b. f x g 2 x g 3 2 2 x 2 xg g 2 2 3 Here, we replace x with (x+g) and simplify. 2 x 4 xg 2 g 3 2 2 Continued on the next slide… HAWKES LEARNING SYSTEMS Copyright © 2011 Hawkes Learning Systems. All rights reserved. math courseware specialists Example 11: Functional Notation and Function Evaluation (cont.) Given the function f x 2 x 2 3, evaluate: c. f x g f x Notice that the first term of the numerator is the function we just simplified in b. g 2 x 4 xg 2 g 3 2 x 3 2 2 g 2 4 xg 2 g g 4x 2g 2 HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Implied Domain of a Function The domain of the function is implied by the formula used in defining the function. It is assumed that the domain of the function consists of all real numbers at which the function can be evaluated to obtain a real number: any values for the argument that result in division by zero or an even root of a negative number must be excluded from the domain. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 12: Implied Domain of a Function Determine the domain of the following functions. a. f x 3 x 7 x Domain: x 7 or , 7 b. g x x5 x 4 2 To avoid an even root of a negative number we need to find x such that 7 – x ≥ 0. Solving this inequality gives us the domain. We want to avoid any x such that x 2 4 0 , as this would give us a denominator of zero. After solving x 2 4 0 we get x 2 x 2 0 , so we define the domain as excluding 2 and – 2. Domain: x 2, 2 or , 2 2, 2 2,