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81. Odd and Even Functions Recall that a function f is odd if f(-x) = -f(x) or even if f(-x) = f(x) for all real x. (a) Show that a polynomial P(x) that contains only odd powers of x is an odd function. (b) Show that a polynomial P(x) that contains only even powers of x is an even function. (c) Show that if a polynomial P(x) contains both odd and even powers of x, then it is neither an odd nor an even function. (d) Express the function P(x) = x' + 6r' – x² – 2x + 5 as the sum of an odd function and an even function.

81. Odd and Even Functions Recall that a function f is odd if f(-x) = -f(x) or even if f(-x) = f(x) for all real x. (a) Show that a polynomial P(x) that contains only odd powers of x is an odd function. (b) Show that a polynomial P(x) that contains only even powers of x is an even function. (c) Show that if a polynomial P(x) contains both odd and even powers of x, then it is neither an odd nor an even function. (d) Express the function P(x) = x' + 6r' – x² – 2x + 5 as the sum of an odd function and an even function.

Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter2: Functions
Section2.6: Transformations Of Functions
Problem 103E
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. Odd and Even Functions Recall that a function f is odd if
f1x2  f1x2 or even if f1x2  f1x2 for all real x.
(a) Show that a polynomial P1x2 that contains only odd
powers of x is an odd function.
(b) Show that a polynomial P1x2 that contains only even
powers of x is an even function.
(c) Show that if a polynomial P1x2 contains both odd and
even powers of x, then it is neither an odd nor an even
function.
(d) Express the function
P1x2  x5  6x3  x2  2x  5
as the sum of an odd function and an even function

81. Odd and Even Functions Recall that a function f is odd if f(-x) = -f(x) or even if f(-x) = f(x) for all real x. (a) Show that a polynomial P(x) that contains only odd powers of x is an odd function. (b) Show that a polynomial P(x) that contains only even powers of x is an even function. (c) Show that if a polynomial P(x) contains both odd and even powers of x, then it is neither an odd nor an even function. (d) Express the function P(x) = x' + 6r' – x² – 2x + 5 as the sum of an odd function and an even function.
Transcribed Image Text:81. Odd and Even Functions Recall that a function f is odd if f(-x) = -f(x) or even if f(-x) = f(x) for all real x. (a) Show that a polynomial P(x) that contains only odd powers of x is an odd function. (b) Show that a polynomial P(x) that contains only even powers of x is an even function. (c) Show that if a polynomial P(x) contains both odd and even powers of x, then it is neither an odd nor an even function. (d) Express the function P(x) = x' + 6r' – x² – 2x + 5 as the sum of an odd function and an even function.
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