Problem 19 - Thomas Calculus

Question

In Exercises \(19-30,\) say whether the function is even, odd, or neither. Give reasons for your answer. $$ f(x)=3 $$

Step-by-Step Solution

Answer

The function \(f(x) = 3\) is even because \(f(-x) = f(x) = 3\).

Step 1: Understand the Definitions

A function is even if for every x in the domain, \(f(-x) = f(x)\). A function is odd if for every x in the domain, \(f(-x) = -f(x)\). If neither condition is satisfied, the function is neither even nor odd.

Step 2: Evaluate f(-x)

Given the function \(f(x) = 3\), calculate \(f(-x)\). Since \(f(x)\) is a constant function:\[ f(-x) = 3 \]This is because the value of \(f(x)\) does not depend on x; it is always 3 regardless of x.

Step 3: Compare f(-x) and f(x)

To determine if the function is even, check if \(f(-x) = f(x)\) using the results:\[ f(-x) = 3 \]\[ f(x) = 3 \]Since \(f(-x) = f(x)\), the function satisfies the condition for being even.

Step 4: Check for Oddness (Not Necessary)

Even though it's only necessary if you're unsure, confirm the function is not odd:The function would be odd if \(f(-x) = -f(x)\). But:\[ f(-x) = 3 \] and \[ -f(x) = -3 \]Since \(f(-x) eq -f(x)\), the function is not odd.

Key Concepts

Constant Function

A constant function is a type of function that always returns the same value, no matter what input it receives. This means that for any value of \( x \), the result of \( f(x) \) will be the same constant.
For example, in the function \( f(x) = 3 \), the output will always be 3 regardless of whether you substitute a positive, negative, or zero in place of \( x \).

• These functions are represented graphically as horizontal lines on a coordinate plane.
• The slope of a constant function is zero, indicating no change in \( y \) with respect to \( x \).
• Constant functions are among the simplest types of mathematical functions, making them an ideal starting point for understanding more complex functions.

Understanding constant functions is essential when learning about even and odd functions, as their evaluation for \( -x \) and \( x \) remains unchanged.

Function Symmetry

Function symmetry relates to the visual or algebraic balance of a function. A function can be either even, odd, or neither, depending on its symmetry properties:

• Even Functions: These have symmetry about the y-axis. Mathematically, if \( f(-x) = f(x) \) for all \( x \) in the function's domain, then the function is classified as even.
• Odd Functions: These functions have symmetric properties with respect to the origin. If \( f(-x) = -f(x) \), the function is considered odd.
• Neither: A function that does not satisfy either of these conditions is neither even nor odd.

Symmetry helps in analyzing how functions behave under transformations like rotations and reflections. For instance, the constant function \( f(x) = 3 \) is even because \( f(-x) \) is equal to \( f(x) \). Recognizing the symmetry of functions simplifies solving complex problems in calculus.

Calculus Definitions

In the realm of calculus, definitions are crucial for distinguishing different types of functions based on their characteristics and behaviors. With constant, even, and odd functions, specific definitions need to be understood:

• Constant Function: As defined earlier, this function’s output doesn’t change with varying input, crucial for identifying its symmetry properties.
• Even and Odd Functions: These definitions involve evaluating the function for both \( x \) and \( -x \). Even functions hold the property \( f(-x) = f(x) \), while odd functions satisfy \( f(-x) = -f(x) \).
• Function Behavior: Recognizing how a function behaves allows for easier integration and differentiation in calculus problems.

Consolidating these definitions helps in developing a comprehensive understanding of functions, aiding in deeper explorations such as integration, differentiation, and analysis in calculus.