Problem 23 - Calculus with Concepts in Calculus

Question

Determine whether \(f\) is an even function, an odd function, or neither. $$ f(x)=e^{x} /\left(e^{2 x}+1\right) $$

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Determine if the Function is Even

A function is even if for all \(x\), \(f(x) = f(-x)\). Substitute \(-x\) into the function and calculate \(f(-x)\): \[ f(-x) = \frac{e^{-x}}{e^{-2x} + 1} \] Now, compare this to \(f(x)\). We have:\[ f(x) = \frac{e^x}{e^{2x} + 1} \]Clearly, \( f(x) eq f(-x) \) as \( e^x eq e^{-x} \) and \( e^{2x} eq e^{-2x} \). Thus, \( f(x) \) is not an even function.

Step 2: Determine if the Function is Odd

A function is odd if for all \(x\), \( f(-x) = -f(x) \). Calculate \(-f(x)\): \[ -f(x) = -\frac{e^x}{e^{2x} + 1} \]Based on \(f(-x)\) from the previous step:\[ f(-x) = \frac{e^{-x}}{e^{-2x} + 1} \]To determine if the function is odd, compare \(f(-x)\) and \(-f(x)\).The expressions \(\frac{e^{-x}}{e^{-2x} + 1}\) and \(-\frac{e^x}{e^{2x} + 1}\) are not equivalent, as both components of the fraction are not straightforward negations or transformations of each other. Thus, \( f(x) \) is not an odd function.

Step 3: Conclusion

Since \( f(x) \) is neither equal to \( f(-x) \) nor \( -f(x) \), the function \( f(x) = \frac{e^x}{e^{2x} + 1} \) is neither even nor odd.

Key Concepts

Function Symmetry

In mathematics, function symmetry plays a crucial role in understanding how functions behave. Knowing whether a function is even, odd, or neither can help simplify complex problems. Let’s break down the symmetries:

• An even function follows the rule: for every point \(x\), \(f(x) = f(-x)\). This creates symmetry about the y-axis. Common examples include \(\cos(x)\).
• An odd function follows: \(f(-x) = -f(x)\), showing symmetry about the origin, like \(\sin(x)\).
• If neither condition holds, the function is neither even nor odd.

Function symmetry can help identify simplifying patterns, which are useful in integration and graph analysis. For instance, if a function is even, areas under its graph over symmetric intervals can cancel out.

Exponential Functions

Exponential functions are extensively used in modeling growth or decay processes. The general form is \(f(x) = e^x\), where \(e\) is a mathematical constant approximately equal to 2.71828. These functions have unique properties including:

• Rapid growth or decay depending on the exponent's sign.
• A unique derivative: \(\frac{d}{dx}e^x = e^x\).

For various applications, it’s important to understand not just the behavior of \(e^x\), but how it interacts with other expressions in complex functions. In the provided solution, the function involves exponential terms in both numerator and denominator, showcasing how they can alter basic function behavior significantly.

Algebraic Manipulation

Algebraic manipulation allows us to transform expressions in ways that reveal important properties or simplify calculations. When determining symmetries, \(f(x)\) in the provided example calls for substitution and comparison:

• Substitute \(-x\)\ into the function to consider possible symmetries.
• Compare \(f(-x)\) to \(f(x)\) and \(-f(x)\)\ to identify even or odd nature.

This technique highlights how expressions can be rewritten or understood better, an essential skill for tackling more complex mathematics problems. Understanding the results of these manipulations assists in drawing conclusions accurately about functions and predicting their graphically observable behaviors.