Problem 19 - The Calculus with Analytic Geometry

Question

Determine whether the composite function \(f \circ g\) is odd or even in each of the following cases: (a) \(f\) and \(g\) are both even; (b) \(f\) and \(g\) are both odd; (c) \(f\) is even and \(g\) is odd; (d) \(f\) is odd and \(g\) is even.

Step-by-Step Solution

Answer

(a) even, (b) odd, (c) even, (d) odd

Step 1 - Recall Definitions

An even function satisfies the property: \( f(x) = f(-x) \) for all \( x \). An odd function satisfies the property: \( f(x) = -f(-x) \) for all \( x \).

Step 2 - Analyze Case (a)

For \( f \) and \( g \) both being even functions: \( f(x) = f(-x) \) and \( g(x) = g(-x) \). Therefore, \( f(g(x)) = f(g(-x)) \), showing that \( f \circ g \) is even.

Step 3 - Analyze Case (b)

For \( f \) and \( g \) both being odd functions: \( f(x) = -f(-x) \) and \( g(x) = -g(-x) \). Therefore, \( f(g(x)) = f(-g(-x)) = -f(g(-x)) \), showing that \( f \circ g \) is odd.

Step 4 - Analyze Case (c)

For \( f \) being even and \( g \) being odd functions: \( f(x) = f(-x) \) and \( g(x) = -g(-x) \). Therefore, \( f(g(x)) = f(-g(-x)) = f(g(-x)) \), showing that \( f \circ g \) is even.

Step 5 - Analyze Case (d)

For \( f \) being odd and \( g \) being even functions: \( f(x) = -f(-x) \) and \( g(x) = g(-x) \). Therefore, \( f(g(x)) = -f(g(-x)) \), showing that \( f \circ g \) is odd.

Key Concepts

Even Functions

Understanding whether a function is even can help in many mathematical problems. An even function is symmetric with respect to the y-axis. That means when you reflect the graph of the function across the y-axis, it remains unchanged. Mathematically, a function is even if it satisfies the condition: An example would be a simple quadratic function and . This property is essential to know, as it can simplify complex problems. It often allows you to consider only non-negative values of x and then use symmetry.

Odd Functions

An odd function behaves differently from an even function. It is symmetric with respect to the origin. This means reflecting the graph of the function across both the x-axis and y-axis leaves the graph unchanged. A function is classified as odd if it satisfies the condition: A common example. would be the cubic function Knowing if a function is odd or even can help in simplifying integrals and solving equations, as it provides valuable symmetry properties

Function Properties

Functions have a variety of important properties beyond just being even or odd. Critical properties include:

Composability: You can combine functions to create new ones using composition Continuity: A function is continuous if you can draw it without lifting your pencil Differentiability: A function is differentiable if it has a derivative at every point in its domain

These properties provide foundational tools for calculus and allow for a deeper understanding of complex mathematical relationships.

Function Composition

Function composition involves creating a new function by applying one function to the results of another. Mathematically, this is written as This operator can help simplify complex functions by breaking them into more manageable parts. In the context of even and odd functions, the composition can change the symmetry properties of the resulting function. For example, if both f and g are even, then Otherwise changes and which can streamline your analysis and computation.