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Chapter 1: Problem 98

Determine whether the statement is true or false. Justify your answer. It is possible for an odd function to have the interval \([0, \infty)\) as its domain.

Short Answer

Expert verified
The statement is false. An odd function cannot have the domain \([0, \infty)\) because the domain of an odd function needs to include both positive and negative values of \( x \) for symmetry about the origin.

Step by step solution

01

Understand the properties of odd functions

An odd function is one where \( f(x) = -f(-x) \) for all \( x \) in the function's domain. This property implies that an odd function has symmetry about the origin.
02

Analyze the given domain

The given domain is \([0, \infty)\), which means the function is defined for all real values from 0 to positive infinity, inclusive of zero.
03

Consider the symmetry of odd functions

To retain the symmetry property about the origin, for any point \( x \) in the domain, there must exist a point \( -x \). However, in the given domain, only non-negative numbers are included.
04

Make a conclusion based on the analysis

Since an odd function requires both positive and negative values of \( x \) for symmetry about the origin, but there are no negative values in the given domain \([0, \infty)\), it's not possible for an odd function to have the domain \([0, \infty)\).

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