SECTION 2.3 Properties of Functions 87 To obtain the graph of a function y f x , ( ) = it is often helpful to know properties of the function and the impact of these properties on the graph of the function. 1 Identify Even and Odd Functions from a Graph The words even and odd , when discussing a function f, describe the symmetry of the graph of the function. A function f is even if and only if, whenever the point x y , ( ) is on the graph of f, the point x y , ( ) − is also on the graph. Using function notation, we define an even function as follows: 2.3 Properties of Functions Now Work the ‘Are You Prepared?’ problems on page 95. • Intervals (Section A.9, pp. A77–A78) • Intercepts (Section 1.3, pp. 20–21) • Slope of a Line (Section 1.5, pp. 32–34) • Point–Slope Form of a Line (Section 1.5, p. 36) • Symmetry (Section 1.3, pp. 21–23) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Identify Even and Odd Functions from a Graph (p. 87) 2 Identify Even and Odd Functions from an Equation (p. 88) 3 Use a Graph to Determine Where a Function Is Increasing, Decreasing, or Constant (p. 89) 4 Use a Graph to Locate Local Maxima and Local Minima (p. 90) 5 Use a Graph to Locate the Absolute Maximum and the Absolute Minimum (p. 91) 6 Use a Graphing Utility to Approximate Local Maxima and Local Minima and to Determine Where a Function Is Increasing or Decreasing (p. 93) 7 Find the Average Rate of Change of a Function (p. 93) Figure 21 (–x1, f(–x1)) (–x2, f(–x2)) (x2, f(x2)) (x1, f(x1)) x y (a) Even function (b) Odd function (x1, f(x1)) (x2, f(x2)) (–x1, f(–x1)) (–x2, f(–x2)) x y DEFINITION Even Function A function f is even if, for every number x in its domain, the number x− is also in the domain and f x f x ( ) ( ) − = DEFINITION Odd Function A function f is odd if, for every number x in its domain, the number x− is also in the domain and f x f x ( ) ( ) − = − For the even function shown in Figure 21(a), notice that f x f x 1 1 ( ) ( ) = − and that f x f x . 2 2 ( ) ( ) = − A function f is odd if and only if, whenever the point x y , ( ) is on the graph of f, the point x y , ( ) − − is also on the graph. Using function notation, we define an odd function as follows: For the odd function shown in Figure 21(b), notice that f x f x , 1 1 ( ) ( ) = − − which is equivalent to f x f x , 1 1 ( ) ( ) − = − and that f x f x , 2 2 ( ) ( ) = − − which is equivalent to f x f x . 2 2 ( ) ( ) − = −
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