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Chapter 6: Problem 85

Suppose \(f\) is a function with period \(p\). Explain why $$ f(x+2 p)=f(x) $$ for every number \(x\) in the domain of \(f\).

Short Answer

Expert verified
A function f is called periodic with period p if for all x in its domain, f(x + p) = f(x). Since f is a function with period p, f(x + p) = f(x). Now for f(x + 2p), we can break this expression as f((x + p) + p). Using the periodic property again, we have f((x + p) + p) = f(x + p) = f(x). Therefore, f(x + 2p) = f(x) for every number x in the domain of f.

Step by step solution

01

Understanding the definition of a periodic function

A function f is called periodic with period p if for all x in its domain, f(x + p) = f(x). This means that the function repeats itself after an interval of p units.
02

Applying the periodic property

We are given that f is a function with period p. Therefore, applying the periodic property, we have: 1. For all x in the domain of f, f(x + p) = f(x) Now let's consider f(x + 2p). We can break down this expression as f((x + p) + p).
03

Using the periodic property again

Since f has period p, we can use the periodic property again on the expression f((x + p) + p) to get: 2. f((x + p) + p) = f(x + p)
04

Comparing the expressions

Now, comparing expressions (1) and (2) from Steps 2 and 3: f((x + p) + p) = f(x + p) = f(x) Therefore, we can conclude that for any function f with period p, f(x + 2p) = f(x) for every number x in the domain of f.

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