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Chapter 1: Problem 122

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
No, the statement is false. An odd function cannot have \([0, \infty)\) as its domain because by definition, an odd function must be symmetric about the origin, and this requires negative values in its domain.

Step by step solution

01

Define an Odd Function

An odd function is a function that satisfies the property \(f(-x) = -f(x)\) for all \(x\) in its domain. If we visualize this, it means that any odd function is symmetric about the point (0,0). In other words, if we fold the graph about the y-axis, it coincides with itself.
02

Reflect Upon the Domain

The domain of a function is the set of all possible input or argument values that the function can accept. It's given in the problem that the domain of the function is \([0, \infty)\). This means the function is defined only for x values greater than or equal to 0.
03

Must Consider Negative Values

It is important to remember that for a function to be odd, it must be defined on both sides of the y-axis as \(f(-x) = -f(x)\) must be true for all \(x\). This means there should be negative values of \(x\), as well as positive values of \(x\), in the domain.
04

Identify Possible or Impossible

Considering the definition of odd function and the given domain \([0, \infty)\), such an odd function cannot exist because the domain doesn't include any negative real numbers. Hence, the statement that 'it is possible for an odd function to have the interval \([0, \infty)\) as its domain' is false.

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