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2. In the terminology, a function f(x) is even when its graph is symmetric with respect to the y-axis, and odd when its graph is symmetric with respect to the origin. Prove that (a) f(x) = a,,xr+. 2n+1 .+a,x' +a,x is an odd function. (b) f(x) = a,x +a,-x-, +.+a,x +a, is an odd function. (c) The product of two even (or two odd) function is even. (d) The product of an odd function and an even function is odd.

2. In the terminology, a function f(x) is even when its graph is symmetric with respect to the y-axis, and odd when its graph is symmetric with respect to the origin. Prove that (a) f(x) = a,,xr+. 2n+1 .+a,x' +a,x is an odd function. (b) f(x) = a,x +a,-x-, +.+a,x +a, is an odd function. (c) The product of two even (or two odd) function is even. (d) The product of an odd function and an even function is odd.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2. In the terminology, a function f(x) is even when its graph is symmetric with respect to the y-axis, and odd when its graph is symmetric with respect to the origin. Prove that 20+1 (a) f(x) = anex +..+a,x' + a,x is an odd function. (b) f(x) = a,,x" +a,n-2x 2n-2 .+a,x' +a, is an odd function. +..... (c) The product of two even (or two odd) function is even. (d) The product of an odd function and an even function is odd.
Transcribed Image Text:2. In the terminology, a function f(x) is even when its graph is symmetric with respect to the y-axis, and odd when its graph is symmetric with respect to the origin. Prove that 20+1 (a) f(x) = anex +..+a,x' + a,x is an odd function. (b) f(x) = a,,x" +a,n-2x 2n-2 .+a,x' +a, is an odd function. +..... (c) The product of two even (or two odd) function is even. (d) The product of an odd function and an even function is odd.
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