Problem 17 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether the function is even, odd, or neither. (a) \(f(x)=\frac{1}{x^{2}}\) (b) \(g(x)=\frac{2}{x^{3}}\)

Step-by-Step Solution

Answer

The function \(f(x)=\frac{1}{x^{2}}\) is even and the function \(g(x)=\frac{2}{x^{3}}\) is odd.

Step 1: Checking If Function (a) Is Even or Odd

To check if \(f(x)=\frac{1}{x^{2}}\) is even or odd, we calculate \(f(-x)\). Substituting \(-x\) in place of \(x\) in the function, we get: \(f(-x)=\frac{1}{(-x)^{2}}= \frac{1}{x^{2}}\). As \(f(x)=f(-x)\), the function \(f(x)=\frac{1}{x^{2}}\) is 'even'.

Step 2: Checking If Function (b) Is Even or Odd

Now we need to check function \(g(x)=\frac{2}{x^{3}}\). Similarly, we substitute \(-x\) in place of \(x\) in \(g(x)\): \(g(-x)=\frac{2}{(-x)^{3}}= -\frac{2}{x^{3}}\). As this equals to \(-g(x)\), the function \(g(x)=\frac{2}{x^{3}}\) is 'odd'.

Key Concepts

Even and Odd Functions

Understanding whether a function is even, odd, or neither is fundamental in calculus. An even function is characterized by symmetry with respect to the y-axis, meaning that for every point \( (x, y) \) on the function, there is a corresponding point \( (-x, y) \) that also lies on the function. Formally, a function \( f(x) \) is even if \( f(x) = f(-x) \) for all values of \( x \). A real-world example would be the parabola \( y = x^2 \), which displays this type of symmetry.

Conversely, an odd function has rotational symmetry about the origin, which means that rotating the function 180 degrees around the origin leaves it unchanged. Mathematically, a function \( g(x) \) is odd if \( g(-x) = -g(x) \) for all values of \( x \). The classic example of an odd function is \( y = x^3 \). In our exercise, \( f(x)=\frac{1}{x^{2}} \) is even because it remains unchanged when \( x \) is replaced by \( -x \). On the other hand, \( g(x)=\frac{2}{x^{3}} \) is odd since substituting \( x \) with \( -x \) yields the negative of the original function.

Symmetry in Calculus

The concept of symmetry in calculus is instrumental in understanding the behavior of functions and can greatly simplify the process of solving problems. Symmetry can reveal crucial information about a function's limits, integrals, and derivatives. For example, if a function is even, the area under the curve from \( -a \) to \( a \) is simply twice the area from \( 0 \) to \( a \) due to the symmetrical nature of the graph.

Symmetry can also affect the function's Taylor series expansion; even functions will only have cosine and even polynomial terms, while odd functions will only have sine and odd polynomial terms. This distinction can be used to predict the function's behavior and solve calculus problems more efficiently. In summary, identifying symmetry helps in visualizing the function and often leads to straightforward calculations in many calculus operations.

Rational Functions

Rational functions are ratios of polynomials, which means they take the form \( \frac{p(x)}{q(x)} \) where \( p(x) \) and \( q(x) \) are polynomials. These functions can exhibit symmetrical properties and can either be even or odd. They are versatile in modeling real-world scenarios, such as rates of change, and are often subject to analysis in calculus to understand their asymptotic behavior and discontinuities.

A key characteristic of rational functions is their vertical asymptotes, which occur at the zeroes of the denominator, provided these points are not also zeroes of the numerator. They also may have horizontal asymptotes, depending on the degrees of the numerator and denominator polynomials. Functions \( f(x) \) and \( g(x) \) from the exercise are examples of rational functions, and analyzing their evenness or oddness is part of understanding their broader behavior within the realm of calculus.