Problem 78 - Thomas Calculus

Question

Can a function be both even and odd? Give reasons for your answer.

Step-by-Step Solution

Answer

A function can be both even and odd only if it is the zero function.

Step 1: Understand the Definitions

An even function satisfies the condition \(f(x) = f(-x)\) for all values in its domain. An odd function satisfies \(f(-x) = -f(x)\) for its domain. These definitions will guide us in determining if a function can be both even and odd.

Step 2: Analyze Simultaneous Equations

If a function is both even and odd, then it must satisfy both conditions: \(f(x) = f(-x)\) and \(f(-x) = -f(x)\). Setting these equations equal due to the same argument \(f(-x)\), equate \(f(-x) = f(x)\) to \(f(-x) = -f(x)\).

Step 3: Solve the Equations

From the two conditions, equating \(f(x) = -f(x)\), we find that \(2f(x) = 0\). This simplifies to \(f(x) = 0\), meaning the only function that satisfies both conditions is \(f(x) = 0\) for all \(x\).

Step 4: Conclude with Explanation

Since the only function that can be both even and odd is the trivial zero function, no non-zero function can be both even and odd. Therefore, in general, a function cannot be both even and odd unless it is identically zero.

Key Concepts

Function Properties

Functions have distinct properties that help to classify and understand their behavior. These properties are important to grasp as they dictate how a function responds to inputs within its domain.

To categorize functions, we look at symmetries. Namely, even and odd functions possess unique symmetries:

• Even Function: Symmetric with respect to the y-axis. The graph of the function will look the same when you fold it along the y-axis.
• Odd Function: Symmetric with respect to the origin. If you rotate the graph 180 degrees around the origin, the graph appears unchanged.

Understanding these properties helps predict and calculate function behavior efficiently.

Even Function Definition

An even function has a special characteristic, captured by the equation \( f(x) = f(-x) \).

This means for every input 'x,' the output of the function will be the same as the output for '-x.'

Graphically, this reflects as symmetry along the y-axis. Common examples of even functions include \( f(x) = x^2 \) and \( f(x) = \cos x \). These functions will display a mirror-like symmetry, providing a helpful way to visualize and verify their even nature in practical scenarios.

Odd Function Definition

Odd functions, on the other hand, adhere to the condition \( f(-x) = -f(x) \).

This implies that flipping the sign of your input results in the negation of the output. So if you substitute '-x' into the function, you should achieve the negative of what you get with 'x.'

In terms of visualization, odd functions exhibit rotational symmetry about the origin. Examples of odd functions include \( f(x) = x^3 \) and \( f(x) = \sin x \). Testing these equations can solidify understanding and assist in identifying when functions exhibit these characteristics.