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Magazine Chapter 1: Problem 77
Assume that \(f\) is an even function, \(g\) is an odd function, and both \(f\) and \(g\) are defined on the entire real line \((-\infty, \infty) .\) Which of the following (where defined) are even? odd? foLLowing a. \(f g\) b. \(f / g\) c. \(g / f\) d. \(f^{2}=f f\) e. \(g^{2}=g g\) f. \(f^{\circ} g\) g. \(g \circ f\) h. \(f \circ f\) i. \(g^{\circ} \mathrm{g}\)
Short Answer
Expert verified
a. odd, b. odd, c. odd, d. even, e. even, f. even, g. odd, h. even, i. odd.
Step by step solution
01
Review Definitions
First, recall that an even function satisfies the property \( f(x) = f(-x) \) for all \( x \) in its domain. An odd function satisfies \( g(x) = -g(-x) \) for all \( x \). These properties are essential for analyzing each part of the problem.
02
Analyze Product of Functions \( f \) and \( g \)
Consider \( fg(x) = f(x)g(x) \). Substitute \( x \) with \( -x \) to get \( fg(-x) = f(-x)g(-x) = f(x)(-g(x)) = -f(x)g(x) = -fg(x) \). Since \( fg(-x) = -fg(x) \), the product \( fg \) is an odd function.
03
Analyze Quotient \( f/g \)
Consider \( \frac{f}{g}(x) = \frac{f(x)}{g(x)} \). Substitute \( x \) with \( -x \) to get \( \frac{f}{g}(-x) = \frac{f(-x)}{g(-x)} = \frac{f(x)}{-g(x)} = -\frac{f(x)}{g(x)} = -\frac{f}{g}(x) \). This shows \( \frac{f}{g} \) is odd.
04
Analyze Quotient \( g/f \)
Consider \( \frac{g}{f}(x) = \frac{g(x)}{f(x)} \). Substitute \( x \) with \( -x \) to get \( \frac{g}{f}(-x) = \frac{g(-x)}{f(-x)} = \frac{-g(x)}{f(x)} = -\frac{g(x)}{f(x)} = -\frac{g}{f}(x) \). This shows \( \frac{g}{f} \) is odd.
05
Analyze Square \( f^2(x) = ff(x) \)
Since \( f(x) \) is even, substitute \( x \) with \( -x \): \( f^2(-x) = f(-x)f(-x) = f(x)f(x) = f^2(x) \). Thus, \( f^2 \) is an even function.
06
Analyze Square \( g^2(x) = gg(x) \)
Since \( g(x) \) is odd, substitute \( x \) with \( -x \): \( g^2(-x) = g(-x)g(-x) = (-g(x))(-g(x)) = g(x)g(x) = g^2(x) \). Thus, \( g^2 \) is an even function.
07
Analyze Composition \( f \circ g \)
Consider \((f \circ g)(x) = f(g(x))\). Since \( g(x) \) is odd, \( g(-x) = -g(x) \), substituting gives \((f \circ g)(-x) = f(g(-x)) = f(-g(x)) = f(g(x)) = (f \circ g)(x) \). So \( f \circ g \) is an even function.
08
Analyze Composition \( g \circ f \)
Consider \((g \circ f)(x) = g(f(x))\). Since \( f(x) \) is even, \( f(-x) = f(x) \), substituting gives \((g \circ f)(-x) = g(f(-x)) = g(f(x)) = -g(f(x)) = -(g \circ f)(x) \). So \( g \circ f \) is an odd function.
09
Analyze Composition \( f \circ f \)
Consider \((f \circ f)(x) = f(f(x))\). Since \( f \) is even, \( f(-x) = f(x) \), substituting gives \((f \circ f)(-x) = f(f(-x)) = f(f(x)) = (f \circ f)(x) \). So \( f \circ f \) is an even function.
10
Analyze Composition \( g \circ g \)
Consider \((g \circ g)(x) = g(g(x))\). Since \( g \) is odd, \( g(-x) = -g(x) \), substituting gives \(g(g(-x)) = g(-g(x)) = -g(g(x)) = -(g \circ g)(x)\). So \( g \circ g \) is an odd function.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Function Composition
Function composition is an operation that takes two functions, say, \( f \) and \( g \), and produces a new function. This new function, denoted as \( f \circ g \), is defined as \( (f \circ g)(x) = f(g(x)) \). It's like putting one function inside another function.
- Composition is not commutative, meaning \( f \circ g \) is generally not equal to \( g \circ f \).
- For two functions to be composed, the output of one function (the range) must be compatible with the input of the other function (the domain).
Properties of Functions
Functions have various properties that categorize them, including the classification as even or odd functions. These properties greatly influence their composition and algebraic capabilities.
- **Even Functions:** These functions satisfy the property \( f(x) = f(-x) \). Graphically, they are symmetrical with respect to the y-axis, meaning they're mirrored.
- Example: The function \( f(x) = x^2 \) is even. - **Odd Functions:** These satisfy the property \( g(x) = -g(-x) \). Odd functions have rotational symmetry about the origin.
- Example: The function \( g(x) = x^3 \) is odd.
Function Algebra
Function algebra involves operations such as addition, subtraction, multiplication, and division of functions, much like algebraic operations with numbers.
- **Multiplication:** If you multiply an even function with an odd function, like in the case of \( f(x)g(x) \), the product is often odd.
- **Division:** The quotient of an even function by an odd function is generally odd, as you can see in expressions like \( \frac{f(x)}{g(x)} \).
- **Powers:** When an even function is raised to any power, the result tends to remain even. Conversely, squaring an odd function (like \( g^2(x) \)) results in an even function. This is because \((-1)^2 = 1\).
