Problem 7 - Basic Technical Mathematics with Calculus

Question

Find the domain and range of the given functions. In Exercises 11 and \(12,\) explain your answers. $$G(R)=\frac{3.2}{R}$$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Function

The given function is \( G(R) = \frac{3.2}{R} \). This is a rational function where the numerator is a constant \( 3.2 \), and the denominator is the variable \( R \). Our goal is to find the domain and range of this function.

Step 2: Determine the Domain

The domain of a function is the set of all possible input values (\( R \) values) for which the function is defined. Since division by zero is undefined, we need to exclude \( R = 0 \) from the domain. Therefore, the domain is all real numbers except \( R = 0 \). In interval notation, this is expressed as \( (-\infty, 0) \cup (0, \infty) \).

Step 3: Determine the Range

The range of a function is the set of all possible output values (\( G(R) \) values). Since the function \( G(R) = \frac{3.2}{R} \) can take any positive or negative real number except zero (because the output is also undefined when \( R \) approaches zero), the outputs also cannot be zero. Thus, the range is all real numbers except zero, expressed as \( (-\infty, 0) \cup (0, \infty) \).

Step 4: Summarize the Results

Based on the determinations above: - The domain of \( G(R) \) is \( (-\infty, 0) \cup (0, \infty) \). - The range of \( G(R) \) is \( (-\infty, 0) \cup (0, \infty) \).

Key Concepts

Rational Functions

A rational function is a type of function represented by the ratio of two polynomials, where the polynomial in the numerator and the one in the denominator can include constants or variables. It's essential to understand that the expression is in the form \[ f(x) = \frac{P(x)}{Q(x)} \]where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \) to avoid undefined values.

Rational functions are versatile and can model many real-world situations, such as rates and proportions. For our function, \( G(R) = \frac{3.2}{R} \), we see that this specific form is simple because the numerator is a constant value, 3.2, and the denominator is the variable \( R \). This basic structure is critical, as it simplifies our understanding of how inputs affect outputs.

• Numerator: The top part of the fraction (in this case, \( 3.2 \)).
• Denominator: The bottom part of the fraction (\( R \), which varies).

Understanding rational functions help us predict behavior and constraints when dealing with different values of \( R \).

Division by Zero

When dealing with rational functions like \( G(R) = \frac{3.2}{R} \), division by zero is a crucial consideration. At any point, if \( R = 0 \), the denominator becomes zero, resulting in an undefined function. This is because dividing any number by zero doesn't yield a meaningful or finite number. Here's why:

• Division by zero is undefined in mathematics; no number multiplied by zero can ever result in a value different than zero.
• This leads to a violation of fundamental arithmetic rules when attempting such division.

Thus, for rational functions, values that cause the denominator to be zero must be excluded from the domain. For \( G(R) \), \( R = 0 \) makes the function undefined, so we remove it from possible input values. This exclusion ensures our function behavior is consistent and mathematically valid.

Interval Notation

Interval notation is a compact way of describing certain continuous subsets of real numbers. When finding the domain and range of functions like \( G(R) = \frac{3.2}{R} \), we use interval notation to express the range of possible numbers without writing them explicitly.

To specify intervals, parentheses \(( )\) are used to denote that an endpoint is not included, while brackets \([ ]\) include an endpoint. For \( G(R) \):

• Domain: \((-\infty, 0) \cup (0, \infty) \) means the function includes all real numbers except where \( R = 0 \).
• Range: Exactly the same as the domain, indicating outputs can be any number other than zero.

The union symbol \(\cup\) combines multiple intervals, reflecting how our function splits the real line around excluded points. In practice, interval notation succinctly expresses complicated concepts, ensuring a clear understanding of variable restrictions and function outputs.