Problem 67 Suppose a function has the prope... [FREE SOLUTION]

Chapter 0: Problem 67

Suppose a function has the property that whenever \(x\) is in the domain of \(f\), then so is \(-x .\) Show that \(f\) can be written as the sum of an even function and an odd function.

Short Answer

Expert verified

Given that whenever x is in the domain of f, -x is also in the domain of f, we can define even function g(x) and odd function h(x) as: g(x) = \(\frac{1}{2}[f(x) + f(-x)]\) h(x) = \(\frac{1}{2}[f(x) - f(-x)]\) Then, we can write f(x) as the sum of g(x) and h(x): f(x) = g(x) + h(x) Thus, we have shown that the function f can be written as the sum of an even function and an odd function.

Step by step solution

Recall the definitions of even and odd functions

An even function is a function that satisfies the condition f(x) = f(-x) for all x in its domain. An odd function is a function that satisfies the condition f(x) = -f(-x) for all x in its domain.

Find general formulas for even and odd functions

Let g(x) be an even function and h(x) be an odd function. Then, we can define them as follows: g(x) = g(-x) for all x in its domain h(x) = -h(-x) for all x in its domain

Write f(x) as the sum of an even function and an odd function

We will write f(x) as the sum of g(x) and h(x), which are the even and odd functions respectively. The goal is to find expressions for g(x) and h(x) in terms of f(x). f(x) = g(x) + h(x)

Find expressions for g(x) and h(x) in terms of f(x)

To find the expressions for g(x) and h(x) in terms of f(x), we can use the properties of even and odd functions. First, we write the even function g(x) as the average of f(x) and f(-x): g(x) = \(\frac{1}{2}[f(x) + f(-x)]\) Next, we write the odd function h(x) as the average of the difference between f(x) and f(-x): h(x) = \(\frac{1}{2}[f(x) - f(-x)]\)

Rewrite f(x) in terms of g(x) and h(x)

Now, we can rewrite f(x) as the sum of g(x) and h(x) using the expressions derived in the previous step: f(x) = \(\frac{1}{2}[f(x) + f(-x)] + \frac{1}{2}[f(x) - f(-x)]\) f(x) = g(x) + h(x) Thus, we have shown that the function f can be written as the sum of an even function and an odd function.

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91影视

Suppose a function has the property that whenever \(x\) is in the domain of \(f\), then so is \(-x .\) Show that \(f\) can be written as the sum of an even function and an odd function.

Short Answer

Step by step solution

Recall the definitions of even and odd functions

Find general formulas for even and odd functions

Write f(x) as the sum of an even function and an odd function

Find expressions for g(x) and h(x) in terms of f(x)

Rewrite f(x) in terms of g(x) and h(x)

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