A function f is even if f(x) = f(-x) and odd if f(x) = -f(-x). For example, f is even if f(x) = x2 or f(x) = |x| or f(x) = cosx, while f is odd if f(x) = x or f(x) = sinx. (a) Determine whether f + g is even, odd, or not necessarily either, in the four cases obtained by choosing f even or odd, and g even or odd. (Your answers can most conveniently be displayed in a 2 x 2 table.) (b) Do the same for f • g. (c) Do the same for f o g. (d) Prove that every even function f can be written f(x) = g(|x|), for infinitely many functions g.
A function f is even if f(x) = f(-x) and odd if f(x) = -f(-x). For example, f is even if f(x) = x2 or f(x) = |x| or f(x) = cosx, while f is odd if f(x) = x or f(x) = sinx. (a) Determine whether f + g is even, odd, or not necessarily either, in the four cases obtained by choosing f even or odd, and g even or odd. (Your answers can most conveniently be displayed in a 2 x 2 table.) (b) Do the same for f • g. (c) Do the same for f o g. (d) Prove that every even function f can be written f(x) = g(|x|), for infinitely many functions g.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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A function f is even if f(x) = f(-x) and odd if f(x) = -f(-x). For
example, f is even if f(x) = x2 or f(x) = |x| or f(x) = cosx, while f is
odd if f(x) = x or f(x) = sinx.
(a) Determine whether f + g is even, odd, or not necessarily either, in the
four cases obtained by choosing f even or odd, and g even or odd. (Your
answers can most conveniently be displayed in a 2 x 2 table.)
(b) Do the same for f • g.
(c) Do the same for f o g.
(d) Prove that every even function f can be written f(x) = g(|x|), for infinitely
many functions g.
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