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Chapter 1: Problem 68

Suppose \(f\) and \(g\) are both odd functions. Is the composition \(f \circ g\) even, odd, or neither? Explain.

Short Answer

Expert verified
The composition \(f \circ g\) is an odd function, since \((f\circ g)(-\text{x}) = -f(g(\text{x})) = -(f\circ g)(\text{x})\), which satisfies the definition of an odd function.

Step by step solution

01

Recall the definition of even and odd functions

An even function satisfies the property \(f(-x) = f(x)\) for all \(x\) in its domain, while an odd function satisfies the property \(f(-x) = -f(x)\) for all \(x\) in its domain.
02

Determine the behavior of the composite function at -x

We want to analyze the composite function, \(f\circ g\), at the input \(-x\). This means we need to calculate \((f\circ g)(-\text{x})\), which is equal to \(f(g(-\text{x}))\).
03

Apply the properties of odd functions to g(-x)

Since \(g\) is an odd function, we know that \(g(-\text{x}) = -g(\text{x})\). Thus, we can rewrite the composite function as: \[ f(g(-\text{x})) = f(-g(\text{x})). \]
04

Apply the properties of odd functions to f(-g(x))

Since \(f\) is also an odd function, we know that \(f(-y) = -f(y)\) for all \(y\) in the domain of \(f\). In this case, we can let \(y = g(\text{x})\), so we have the following relationship: \[ f(-g(\text{x})) = -f(g(\text{x})). \]
05

Conclude the nature of the composite function

Combining our results from Steps 3 and 4, we have \((f\circ g)(-\text{x}) = -f(g(\text{x})) = -(f\circ g)(\text{x})\). This means that the composite function \(f\circ g\) is an odd function since it satisfies the definition of an odd function.

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