SECTION 2.3 Properties of Functions 89 3 Use a Graph to Determine Where a Function Is Increasing, Decreasing, or Constant Consider the graph given in Figure 26. If you look from left to right along the graph of the function, you will notice that parts of the graph are going up, parts are going down, and parts are horizontal. In such cases, the function is described as increasing, decreasing , or constant , respectively. (b) Graph the function. See Figure 24. It appears that there is no symmetry. We conjecture that the function is neither even nor odd. To algebraically verify the conjecture that the function is not even, find g x ( ) − and compare the result with g x . ( ) g x x x g x x 1 1; 1 3 3 3 ( ) ( ) ( ) − = − − = − − = − Since g x g x , ( ) ( ) − ≠ the function is not even. To algebraically verify the conjecture that the function is not odd, find g x( ) − and compare the result with g x . ( ) − g x x x g x x 1 1; 1 3 3 3 ( ) ( ) ( ) − =− − =−+ −=−− Since g x g x , ( ) ( ) − ≠ − the function is not odd. The graph is not symmetric with respect to the y -axis nor is it symmetric with respect to the origin. (c) Graph the function. See Figure 25. It appears that there is symmetry with respect to the origin. We conjecture that the function is odd. To algebraically verify the conjecture that the function is odd, replace x by x− in h x x x 5 . 3 ( ) = − Then h x x x x x x x h x 5 5 5 3 3 3 ( ) ( ) ( ) ( ) ( ) −=− −−=− +=− − =− Since h x h x , ( ) ( ) − = − h is an odd function and the graph of h is symmetric with respect to the origin. Figure 24 10 210 23 3 Figure 25 3 23 22 2 Now Work PROBLEM 37 In Words • If a function is increasing, then as the values of x get bigger, the values of the function also get bigger. • If a function is decreasing, then as the values of x get bigger, the values of the function get smaller. • If a function is constant, then as the values of x get bigger, the values of the function remain unchanged. Figure 26 x y 6 5 22 24 (3, 4) (6, 1) (0, 4) (22, 0) (26, 0) (24, 22) y = f(x) DEFINITIONS Increasing Function, Decreasing Function, Constant Function • A function f is increasing on an interval I if, for any choice of x1 and x2 in I, with x x , 1 2 < then f x f x . 1 2 ( ) ( ) < • A function f is decreasing on an interval I if, for any choice of x1 and x2 in I, with x x , 1 2 < then f x f x . 1 2 ( ) ( ) > • A function f is constant on an interval I if, for all choices of x in I, the values f x( ) are equal. Need to Review? Interval Notation is discussed in Section A.9, p. A77. Figure 27 on the next page illustrates the definitions. Note that we always read a graph from left to right because of the statement x x 1 2 < in the definition.The graph of an increasing function goes up from left to right, the graph of a decreasing function goes down from left to right, and the graph of a constant function remains at a fixed
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