Problem 9 - Basic Technical Mathematics with Calculus

Question

Find the domain and range of the given functions. In Exercises 11 and \(12,\) explain your answers. $$f(s)=\frac{2}{s^{2}}$$

Step-by-Step Solution

Answer

Domain: \((-\infty, 0) \cup (0, \infty)\); Range: \((0, \infty)\).

Step 1: Understanding the Function

The given function is \( f(s) = \frac{2}{s^2} \). This is a rational function with a numerator of 2 and a denominator of \( s^2 \). Our goal is to find the domain and range.

Step 2: Finding the Domain

The domain of a function is the set of all possible values of \( s \) for which the function is defined. For \( f(s) = \frac{2}{s^2} \), the function is undefined where the denominator equals zero. Thus, we set \( s^2 = 0 \) and solve, giving \( s = 0 \). The domain of the function is all real numbers except \( s = 0 \). In interval notation, the domain is \((-\infty, 0) \cup (0, \infty)\).

Step 3: Understanding the Range

The range of a function is the set of all possible output values (\( f(s) \)). Since \( s^2 \) is always positive for \( s eq 0 \), \( \frac{2}{s^2} \) is also positive. As \( s \) approaches zero from either direction, \( \frac{2}{s^2} \) approaches infinity. Conversely, as \( s \) moves away from zero, \( \frac{2}{s^2} \) approaches zero. Thus, the range of \( f(s) \) is \((0, \infty)\).

Step 4: Explanation of Domain and Range

The domain excludes \( s = 0 \) because the function becomes undefined. The range excludes zero because \( \frac{2}{s^2} \) can never be zero, as \( 2 \) can never be divided by a nonzero number to get zero, and \( s^2 \) is always positive.

Key Concepts

Rational Functions

Rational functions are mathematical expressions representing the ratio of two polynomials. The general form is \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). These types of functions are significant in calculus due to their interesting properties related to limits, asymptotes, and behavior at various points.Key characteristics of rational functions include:

• They can exhibit vertical or horizontal asymptotes.
• Understanding the domain is critical since any value of \( x \) that makes \( Q(x) = 0 \) leads to an undefined function.

In our example, \( f(s) = \frac{2}{s^2} \), the numerator is 2, and the denominator is \( s^2 \). Since \( s^2 \) cannot be zero, the point where \( s = 0 \) must be excluded from the domain.

Domain and Range

The domain of a rational function is concerned with the permissible input values that do not make the denominator zero.For \( f(s) = \frac{2}{s^2} \), the denominator is \( s^2 \). Solving \( s^2 = 0 \) gives \( s = 0 \), indicating that the function is undefined at this point. Thus, the domain excludes this value and includes all other real numbers.In interval notation, the domain is expressed as

• \((-\infty, 0) \cup (0, \infty)\)

For the range, we look at the possible output values. Since \( 2 \div s^2 \) is always positive for \( s eq 0 \) and can never equal zero, we conclude that the range of \( f(s) \) is

• \((0, \infty)\)

Function Analysis

Analyzing a function involves understanding its behavior, limits, and continuity.- **Undefined Points**: For \( f(s) = \frac{2}{s^2} \), the function is undefined at \( s = 0 \). This is because division by zero is undefined.- **Behavior Near Undefined Points**: As \( s \to 0 \) from either direction, \( f(s) \to \infty \). This behavior indicates a vertical asymptote at \( s = 0 \).Arithmetic operations of rational functions need attention, especially when determining limits and asymptotic behavior. Observing these characteristics helps students navigate complex corridors of calculus with ease.

Interval Notation

Interval notation is a compact way of describing a set of numbers. It's often used to define domains and ranges of functions in calculus.Intervals use parentheses \(()\) to indicate that endpoints are not included in the set, and brackets \([]\) if they are included. For example:

• \((a, b)\) means every number between \(a\) and \(b\), *excluding* \(a\) and \(b\).
• \([a, b]\) includes both endpoints.
• \((a, b]\) includes \(b\) but not \(a\), and vice versa for \([a, b)\).

The domain of \( f(s) = \frac{2}{s^2} \) is \((-\infty, 0) \cup (0, \infty) \), expressing all real numbers except zero. By mastering interval notation, students efficiently communicate complex mathematical ideas.