285 2.7 Graphing Techniques Even and Odd Functions The concepts of symmetry with respect to the y-axis and symmetry with respect to the origin are closely associated with the concepts of even and odd functions. Even and Odd Functions A function ƒ is an even function if ƒ1 −x2 =ƒ1x2 for all x in the domain of ƒ. (Its graph is symmetric with respect to the y-axis.) A function ƒ is an odd function if ƒ1 −x2 = −ƒ1x2 for all x in the domain of ƒ. (Its graph is symmetric with respect to the origin.) The function ƒ is odd because ƒ1-x2 = -ƒ1x2. (c) ƒ1x2 = 3x2 + 5x ƒ1-x2 = 31-x22 + 51-x2 Replace x with -x. ƒ1-x2 = 3x2 - 5x Simplify. Because ƒ1-x2 ≠ƒ1x2 and ƒ1-x2 ≠-ƒ1x2, the function ƒ is neither even nor odd. S Now Try Exercises 57, 59, and 61. EXAMPLE 5 Determining Whether Functions Are Even, Odd, or Neither Determine whether each function defined is even, odd, or neither. (a) ƒ1x2 = 8x4 - 3x2 + 1 (b) ƒ1x2 = 6x3 - 9x (c) ƒ1x2 = 3x2 + 5x SOLUTION (a) Replacing x with -x gives the following. ƒ1x2 = 8x4 - 3x2 + 1 ƒ1-x2 = 81-x24 - 31-x22 + 1 Replace x with -x. ƒ1-x2 = 8x4 - 3x2 + 1 Apply the exponents. ƒ1-x2 = ƒ1x2 8x4 - 3x2 + 1 = ƒ1x2 Because ƒ1-x2 = ƒ1x2 for each x in the domain of the function, ƒ is even. (b) ƒ1x2 = 6x3 - 9x ƒ1-x2 = 61-x23 - 91-x2 Replace x with -x. ƒ1-x2 = -6x3 + 9x ƒ1-x2 = -ƒ1x2 -6x3 + 9x = -16x3 - 9x2 = -ƒ1x2 Be careful with signs. NOTE Consider a function defined by a polynomial in x. • If the function has only even exponents on x (including the case of a constant where x0 is understood to have the even exponent 0), it will always be an even function. • Similarly, if only odd exponents appear on x, the function will be an odd function.
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