Problem 18 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether the function is even, odd, or neither. (a) \(f(x)=x+\frac{1}{x}\) (b) \(g(x)=1+\frac{1}{x}\)

Step-by-Step Solution

Answer

The function (a) \(f(x) = x + \frac{1}{x}\) and function (b) \(g(x) = 1 + \frac{1}{x}\) are both neither even nor odd.

Step 1: Analyze Function (a)

Plugging \(-x\) into the function \(f(x) = x + \frac{1}{x}\) results in \(f(-x) = -x - \frac{1}{x}\). This doesn't result in the original function (for even) nor the negative of the original function (for odd), therefore the function is neither even nor odd.

Step 2: Analyze Function (b)

Plugging \(-x\) into the function \(g(x) = 1 + \frac{1}{x}\) results in \(g(-x) = 1 - \frac{1}{x}\). This doesn't result in the original function (for even) and it's not the negative of the original function (for odd), so this function is also neither even nor odd.

Key Concepts

Symmetry of Functions

Understanding symmetry in functions is crucial in determining whether a function is even, odd, or neither. Even functions are symmetric about the y-axis, meaning that for every point \( (x, y) \) on the function, there is an identical point \( (-x, y) \). If we take any even function, let's say \( f(x) \), and plug in the opposite of x, which is -x, we would get \( f(-x) = f(x) \). On the other hand, odd functions show symmetry about the origin. This means that if the function \( f(x) \) is odd, then \( f(-x) = -f(x) \) should hold true.

For example, \( f(x) = x^2 \) is an even function because \( f(-x) = (-x)^2 = x^2 \) which is the original function. Conversely, \( g(x) = x^3 \) is an odd function since \( g(-x) = (-x)^3 = -x^3 \) which is the negative of the original function. It's important to test for these symmetries when examining a function's behavior because it can simplify understanding and working with the function in many mathematical contexts.

Algebraic Functions

Algebraic functions are formed by applying algebraic operations, such as addition, subtraction, multiplication, division, and taking roots to polynomial functions. A polynomial function consists of a sum of terms, each consisting of a variable raised to a non-negative integer power and multiplied by a coefficient.

When studying algebraic functions, such as \( f(x) = x + \frac{1}{x} \), it's apparent that this function is neither purely polynomial (due to the term \( \frac{1}{x} \) which is not a non-negative integer power of x) nor is it even or odd since replacing \( x \) with \( -x \) does not result in \( f(-x) \) being equal to \( f(x) \) or \( -f(x) \). Algebraic functions can often hold intricate behaviors that aren't readily seen and require careful analysis.

Rational Functions

Rational functions are a type of algebraic functions represented by the ratio of two polynomials. In other words, they have the form \( \frac{p(x)}{q(x)} \) where both \( p(x) \) and \( q(x) \) are polynomials. One of the main characteristics of rational functions is their potential to have asymptotes, which are lines that the graph of the function approaches but never touches.

Take the given function \( g(x) = 1 + \frac{1}{x} \) as an example. This function has a vertical asymptote at \( x = 0 \) and is neither even nor odd. When analyzing rational functions, particularly for symmetry, it is pivotal to consider the behavior of the function as \( x \) approaches both positive and negative infinity, and as it approaches any vertical asymptotes. This can expose interesting properties about the function's growth and limitations.