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Chapter 0: Problem 98

True or False? 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
False. An odd function cannot have the domain \([0, \infty)\), because the negative part of the domain is required for the symmetry property of odd functions.

Step by step solution

01

Understanding a property of the odd function

Recall that an odd function is defined as a function where \(f(-x) = -f(x)\) for all x in the domain of the function. This means that the function is symmetric with respect to the origin.
02

Assessing the given domain

The domain provided in the exercise is \([0, \infty)\), which includes all non-negative real numbers. If we look at an odd function, the behavior of the function on the positive side of the x-axis is mirrored on the negative side. This is a critical part of understanding. In the given domain, there is no negative side of the x-axis, so there is no 'mirror' for the function's behavior.
03

Concluding the analysis

Given that the formula of the odd function is \(f(-x) = -f(x)\), it's clear that we need both +x and -x in the domain for the equation to hold, which means that the domain must be all real numbers or at least include negative numbers. Because the provided domain \([0, \infty)\) only includes non-negative numbers, there is no room for the 'mirroring' action on the negative side. Hence an odd function cannot have a domain of \([0, \infty)\).

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Most popular questions from this chapter

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\sqrt{4-x^{2}}, \quad 0 \leq x \leq 2 $$

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=x^{5}-2 $$

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ q(x)=(x-5)^{2} $$

In Exercises 33-38, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$ g(x)=(x+5)^{3} $$

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)= \begin{cases}-x, & x \leq 0 \\ x^{2}-3 x, & x>0\end{cases} $$
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