Problem 98 - Algebra and Trigonometry Real Mathematics, Real People

Question

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.

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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 \([0, \infty)\).