Problem 53 - Thomas Calculus

Question

In Exercises \(47-58,\) say whether the function is even, odd, or neither. Give reasons for your answer. $$ g(x)=\frac{1}{x^{2}-1} $$

Step-by-Step Solution

Answer

The function \( g(x) = \frac{1}{x^2 - 1} \) is even.

Step 1: Understand Function Symmetry

A function is defined as even if \( f(-x) = f(x) \) and odd if \( f(-x) = -f(x) \). If neither condition is met, the function is neither even nor odd.

Step 2: Evaluate \( g(-x) \)

To determine the symmetry of the function \( g(x) = \frac{1}{x^2 - 1} \), calculate \( g(-x) \):\[ g(-x) = \frac{1}{(-x)^2 - 1} = \frac{1}{x^2 - 1} = g(x) \]

Step 3: Determine Symmetry Type

Since \( g(-x) = g(x) \), the function \( g(x) \) satisfies the condition for even functions. \( g(x) \) is even.

Key Concepts

Function Symmetry

In mathematics, determining a function's symmetry plays a crucial role in understanding its behavior. Function symmetry refers to how a function may mirror itself in relation to the y-axis or the origin.

There are two main types of function symmetry to consider:

• Even Symmetry: For a function to show even symmetry, it must satisfy the condition \( f(-x) = f(x) \). This means that substituting \(-x\) into the function yields the same output as plugging in \(x\). Such functions are typically symmetric with respect to the y-axis.
• Odd Symmetry: For odd symmetry, the condition \( f(-x) = -f(x) \) must hold. This means that substituting \(-x\) in results in the negation of the value obtained by substituting \(x\). Functions with odd symmetry are symmetric concerning the origin.

Understanding these properties helps in graph sketching and can simplify solving equations.

Even Function

For a function to be classified as even, it must mirror itself perfectly over the y-axis. Generally, if you can "fold" the graph of an even function over the y-axis, it will match its mirror image.

In the original exercise, the function given is \( g(x) = \frac{1}{x^2 - 1} \). To verify if this function is even, we substitute \(-x\) for \(x\) and compare with the original function:

• Calculate \( g(-x) \): \( g(-x) = \frac{1}{(-x)^2 - 1} \)
• Simplify: \( g(-x) = \frac{1}{x^2 - 1} \), which matches \( g(x) \)

Since after substitution the function remains unchanged, it satisfies the condition \( g(-x) = g(x) \). Therefore, \( g(x) \) is an even function. Recognizing even functions can be extremely useful in mathematical analysis and real-world applications as they often imply certain predictable patterns in their graphs.

Symmetry in Mathematics

Symmetry is a fundamental concept in mathematics and has vast applications and significance. At its core, symmetry refers to the balanced proportions and identical arrangements in geometric shapes, equations, or algebraic expressions.

When discussing functions, symmetry provides insights into the graph's shape and can simplify the function's study:

• Graphical Interpretation: Even functions' graphs are mirrored along the y-axis, leading to a more straightforward visualization and often simplifying calculations, such as finding roots or calculating definite integrals over symmetric intervals.
• Practical Applications: Understanding symmetry can aid in diverse fields such as physics for solving problems related to conservation laws, in engineering for designing balanced structures, and in computer science algorithms for optimizing processes.

Symmetry in mathematics not only makes computations easier but also reveals deeper insights into the inherent nature of the problems being investigated.