Section 4: Solving Equations and Inequalities by Graphing

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 2.4 Solving Equations and Inequalities by Graphing 139 Version: Fall 2007 2.4 Solving Equations and Inequalities by Graphing Our emphasis in the chapter has been on functions and the interpretation of their graphs. In this section, we continue in that vein and turn our exploration to the solution of equations and inequalities by graphing. The equations will have the form f ( x ) = g ( x ) , and the inequalities will have form f ( x ) < g ( x ) and/or f ( x ) > g ( x ) . You might wonder why we have failed to mention inequalities having the form f ( x ) g ( x ) and f ( x ) g ( x ) . The reason for this omission is the fact that the solution of the inequality f ( x ) g ( x ) is simply the union of the solutions of f ( x ) = g ( x ) and f ( x ) < g ( x ) . After all, is pronounced “less than or equal.” Similar comments are in order for the inequality f ( x ) g ( x ) . We will begin by comparing the function values of two functions f and g at various values of x in their domains. Comparing Functions Suppose that we evaluate two functions f and g at a particular value of x . One of three outcomes is possible. Either f ( x ) = g ( x ) , or f ( x ) > g ( x ) , or f ( x ) < g ( x ) . It’s pretty straightforward to compare two function values at a particular value if rules are given for each function. l⚏ Example 1. Given f ( x ) = x 2 and g ( x ) = 2 x +3 , compare the functions at x = 2 , 0, and 3. Simple calculations reveal the relations. At x = 2 , f ( 2) = ( 2) 2 = 4 and g ( 2) = 2( 2) + 3 = 1 , so clearly, f ( 2) > g ( 2) . At x = 0 , f (0) = (0) 2 = 0 and g (0) = 2(0) + 3 = 3 , so clearly, f (0) < g (0) . Finally, at x = 3 , f (3) = (3) 2 = 9 and g (3) = 2(3) + 3 = 9 , so clearly, f (3) = g (3) . We can also compare function values at a particular value of x by examining the graphs of the functions. For example, consider the graphs of two functions f and g in Figure 1 . Copyrighted material. See: 1
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140 Chapter 2 Functions Version: Fall 2007 x y f g Figure 1. Each side of the equation f ( x ) = g ( x ) has its own graph. Next, suppose that we draw a dashed vertical line through the point of intersection of the graphs of f and g , then select a value of x that lies to the left of the dashed vertical line, as shown in Figure 2 (a). Because the graph of f lies above the graph of g for all values of x that lie to the left of the dashed vertical line, it will be the case that f ( x ) > g ( x ) for all such x (see Figure 2 (a)). 2 On the other hand, the graph of f lies below the graph of g for all values of x that lie to the right of the dashed vertical line. Hence, for all such x , it will be the case that f ( x ) < g ( x ) (see Figure 2 (b)). 3 x y f g x ( x,f ( x )) ( x,g ( x )) f ( x ) g ( x ) x y f g x ( x,f ( x )) ( x,g ( x )) f ( x ) g ( x ) (a) To the left of the vertical dashed line, the graph of f lies above the graph of g . (b) To the right of the vertical dashed line, the graph of f lies below the graph of g . Figure 2. Comparing f and g .
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