Problem 77 - Thomas Calculus

Question

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? $$ \begin{array}{ll}{\text { a. } f g} & {\text { b. } f / g} & {\text { c. } g / f} \\ {\text { d. } f^{2}=f f} & {\text { e. } g^{2}=g g} & {\text { f. } f \circ g} \\ {\text { g. } g \circ f} & {\text { h. } f \circ f} & {\text { i. } g \circ g}\end{array} $$

Step-by-Step Solution

Answer

a. odd, b. odd, c. odd, d. even, e. even, f. even, g. odd, h. even, i. even.

Step 1: Understanding Even and Odd Functions

An even function satisfies \( f(x) = f(-x) \) for all \( x \). An odd function satisfies \( g(x) = -g(-x) \) for all \( x \). We will use these properties to determine the evenness or oddness of each expression.

Step 2: Analyze Each Expression

Let's analyze each item to check for evenness or oddness.

Step 2a: Analyze fg

\( f(x)g(x) = f(-x)g(-x) = f(x)(-g(x)) = -f(x)g(x) \), hence \( fg \) is odd.

Step 2b: Analyze f/g

\( \frac{f(x)}{g(x)} = \frac{f(-x)}{g(-x)} = \frac{f(x)}{-g(x)} = -\frac{f(x)}{g(x)} \), so \( f/g \) is odd.

Step 2c: Analyze g/f

\( \frac{g(x)}{f(x)} = \frac{g(-x)}{f(-x)} = \frac{-g(x)}{f(x)} = -\frac{g(x)}{f(x)} \), so \( g/f \) is odd.

Step 2d: Analyze f²

\( f(x)^2 = (f(x))^2 = f(-x)^2 = (f(x))^2 \), hence \( f² \) is even.

Step 2e: Analyze g²

\( g(x)^2 = (g(x))^2 = g(-x)^2 = (-g(x))^2 = (g(x))^2 \), so \( g² \) is even.

Step 2f: Analyze f∘g

\( f(g(x)) = f(-g(x)) = f(g(-x)) \) because \( f \) is even and \( g \) is odd. Thus \( f(g(x)) = f(g(-x)) \), so \( f∘g \) is even.

Step 2g: Analyze g∘f

\( g(f(x)) = g(-f(x)) = -g(f(-x)) \) because \( g \) is odd and \( f \) is even. Thus \( g(f(x)) = -g(f(-x)) \), so \( g∘f \) is odd.

Step 2h: Analyze f∘f

\( f(f(x)) = f(f(-x)) \) because \( f \) is even. Therefore, \( f∘f \) is even.

Step 2i: Analyze g∘g

\( g(g(x)) = g(-g(x)) = -g(g(-x)) \). Here, \( g \) is odd twice, making \( g∘g \) even since \(-(-1) = 1\).

Key Concepts

Even function

An even function is one that remains unchanged when the sign of its input is switched. Mathematically, this is expressed as: \[ f(x) = f(-x) \] for all values of \(x\). This symmetry is reflected about the y-axis of a graph. Such functions maintain their shape and output values, regardless of whether they're evaluated at a positive or negative input.

• Common examples include functions like \( f(x) = x^2 \) or \( f(x) = \cos(x) \).
• If you fold the graph of an even function over the y-axis, both sides will align perfectly, marking the characteristic symmetry.
• Even functions are crucial for many mathematical applications where balanced outputs are necessary despite varying inputs.

Odd function

Odd functions have a distinct property: when you swap the input sign, the output becomes the negative of the original. This is described as: \[ g(x) = -g(-x) \] This results in a graph with rotational symmetry at the origin, meaning it looks identical even if rotated 180 degrees around the origin.

• Common examples include \( g(x) = x^3 \) or \( g(x) = \sin(x) \).
• The rotational symmetry means the function will produce mirrored but negatively signed values for inputs \(x\) and \(-x\).
• This quality finds use in physics and engineering, where opposite inputs yield opposite outputs.

Composition of functions

The composition of functions involves applying one function to the result of another function. If we have two functions, \( f \) and \( g \), the composition is written as \( (f \circ g)(x) \), which means apply \( f \) to the output of \( g(x) \). This allows for complex transformations in mathematical analysis and applications.

• For example, if \( f(x) = x^2 \) and \( g(x) = x + 1 \), the composition \( (f \circ g)(x) \) results in \( f(g(x)) = (x+1)^2 \).
• It's essential to follow the order since \((f \circ g)(x)\) is usually different from \( (g \circ f)(x) \).
• Understanding compositions helps in breaking down and analyzing complex systems in mathematics and computing.
• Compositions respect the properties of the individual functions, for instance: \( f \circ g \) of an even function and an odd function results in different symmetry properties, based on how they interact.

Function properties

Functions have various properties that determine their behavior and characteristics, such as being even or odd. These properties aid us in predicting how functions will act under different operations.

• Understanding whether a function is even, odd, or neither can inform you on its graph's symmetry. This is fundamental when analyzing real-world data that can be modeled by these functions.
• Knowing the composition result of functions aids in anticipating the symmetry or anti-symmetry in the composite graphs, a valuable trait in both pure mathematics and applied sciences.
• Analyzing functions through properties like continuity, differentiability, and symmetry can simplify solving more complex algebraic or calculus-based problems by reducing them to known outcomes.

Exploring these properties deepens the understanding of mathematical models, which is essential in fields like economics, engineering, physics, and beyond.