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I need help with the photo I submited. One question from the photo is (1.) An even number plus an odd number, and an even function plus an odd function. I would like it if you would help me understand how to solve all of them becuase I'm completly lost.

I need help with the photo I submited. One question from the photo is (1.) An even number plus an odd number, and an even function plus an odd function. I would like it if you would help me understand how to solve all of them becuase I'm completly lost.

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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I need help with the photo I submited. One question from the photo is (1.) An even number plus an odd number, and an even function plus an odd function. I would like it if you would help me understand how to solve all of them becuase I'm completly lost.

Recall that an even functions satisfies (1) f(-x) = f(x) (2) its graph is symmetric with respect to the y-axis. Recall that an odd functions satisfies (1) f(-x) = –f(x) (2) its graph is symmetric with respect to the origin. One knows that an even number plus an even number is an even number; for example, 2 + 4 = 6. On the other hand, one can prove that an even function plus an even function is an even function; for example, x² + x* is again an even function. The mathematical proof are as follows: (a.) Even numbers. An even number can be written equivalently as 2m for an integer m. Now, given two even numbers 2m and 2n for integers m and n, we have 2m + 2n = 2(m + n), again an integer m + n and thus, again an even number. (b.) Even functions. For two even functions f (x) and g(x), we have (f + g)(x) = f(x) + g(x) so that f + g)(-x) = f(-x) + g(-x) = f (x) + g(x) = (f +g)(x), again an even function. %3D %3D Follow similar argument, investigate that (1.) An even number plus an odd number, and an even function plus an odd function. (2.) An odd number plus an odd number, and an odd function plus an odd function. (3.) An even number times an even number, and an even function tinmes an even function. (4.) An even number times an odd number, and an even function times an odd function. (5.) An odd number times an odd number, and an odd function times an odd function. You should both provide an example and show a mathematical proof, exactly as what has been described for the case (6.) An even number plus an even number, and an even function plus an even function. in the very beginning of this project. Credits won't be granted via pure guess or argument without rigorous mathematical idea. The purpose of this project is to help building a good habit of writing and inspiring an interest in learning mathematics.
Transcribed Image Text:Recall that an even functions satisfies (1) f(-x) = f(x) (2) its graph is symmetric with respect to the y-axis. Recall that an odd functions satisfies (1) f(-x) = –f(x) (2) its graph is symmetric with respect to the origin. One knows that an even number plus an even number is an even number; for example, 2 + 4 = 6. On the other hand, one can prove that an even function plus an even function is an even function; for example, x² + x* is again an even function. The mathematical proof are as follows: (a.) Even numbers. An even number can be written equivalently as 2m for an integer m. Now, given two even numbers 2m and 2n for integers m and n, we have 2m + 2n = 2(m + n), again an integer m + n and thus, again an even number. (b.) Even functions. For two even functions f (x) and g(x), we have (f + g)(x) = f(x) + g(x) so that f + g)(-x) = f(-x) + g(-x) = f (x) + g(x) = (f +g)(x), again an even function. %3D %3D Follow similar argument, investigate that (1.) An even number plus an odd number, and an even function plus an odd function. (2.) An odd number plus an odd number, and an odd function plus an odd function. (3.) An even number times an even number, and an even function tinmes an even function. (4.) An even number times an odd number, and an even function times an odd function. (5.) An odd number times an odd number, and an odd function times an odd function. You should both provide an example and show a mathematical proof, exactly as what has been described for the case (6.) An even number plus an even number, and an even function plus an even function. in the very beginning of this project. Credits won't be granted via pure guess or argument without rigorous mathematical idea. The purpose of this project is to help building a good habit of writing and inspiring an interest in learning mathematics.
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