Problem 29 - Calculus with Concepts in Calculus

Question

Find the domain of the function. $$ g(w)=\frac{2 w-8}{w^{2}-16} $$

Step-by-Step Solution

Answer

The domain of \( g(w) \) is \((-\infty, -4) \cup (-4, 4) \cup (4, \infty)\).

Step 1: Identify the Function's Denominator

The first step in finding the domain of the function is to identify the values of the variable that would make the denominator equal to zero. For the function \( g(w) = \frac{2w - 8}{w^2 - 16} \), the denominator is \( w^2 - 16 \).

Step 2: Set the Denominator to Zero

Set the denominator equal to zero and solve for \( w \):\[w^2 - 16 = 0\]This equation reveals the critical values where the function is undefined.

Step 3: Solve the Equation

Solve the equation \( w^2 - 16 = 0 \). By factoring the equation, we have: \[(w - 4)(w + 4) = 0\]This gives \( w = 4 \) and \( w = -4 \).

Step 4: Determine the Domain

The domain of \( g(w) \) is all real numbers except for those values that make the denominator zero. Thus, the domain of \( g(w) \) is \[w \in \mathbb{R} \setminus \{-4, 4\}\].

Step 5: Express the Domain In Interval Notation

The domain in interval notation excludes the points where the function is undefined: \[(-\infty, -4) \cup (-4, 4) \cup (4, \infty)\].

Key Concepts

Denominator in Functions

In mathematics, the denominator of a fraction represents the bottom part of the fraction. It tells you into how many equal parts the whole is divided. For functions that are rational, like the one in our problem, the denominator is particularly important.

For the function \( g(w) = \frac{2w - 8}{w^2 - 16} \), the denominator is \( w^2 - 16 \). The domain of a function is the set of all possible input values (or \( w \) values here) that keep the function defined. Since dividing by zero is undefined, we seek to find when the denominator becomes zero and exclude such values.

• Identify the denominator: \( w^2 - 16 \)
• Set it equal to zero to find problematic points

This step is crucial to determine where the function is undefined.

Factoring Equations

Factoring is a method used to simplify equations and find solutions. It involves breaking down a complex expression into more manageable components.

In the equation \( w^2 - 16 = 0 \), we can recognize this as a difference of squares. A difference of squares follows the pattern \( a^2 - b^2 = (a - b)(a + b) \).
Here, \( w^2 - 4^2 \) can be factored as:

• \((w - 4)(w + 4) = 0\)

This tells us that the solutions are \( w = 4 \) and \( w = -4 \), indicating points where the denominator (and hence the function) is zero. Factoring is a powerful tool because it transforms a polynomial equation into a product of linear factors, making it much easier to solve.

Using Interval Notation

Interval notation is a mathematical method used to describe a set of numbers along the number line. It is especially useful for conveying the domain of a function.

When determining the domain, we avoid values that make the denominator zero. Since \( w = 4 \) and \( w = -4 \) are these problematic values, we must exclude them. In interval notation, this exclusion can be elegantly expressed as:

• \((-fty, -4) \cup (-4, 4) \cup (4, fty)\)

Here, the union symbol \(\cup\) joins disjoint intervals, effectively saying all real numbers except \( -4 \) and \( 4 \). This provides a clear and concise way to represent continuous intervals of numbers.

Understanding Real Numbers

Real numbers encompass a vast set of numbers that include both rational and irrational numbers. They are essentially all numbers that can be found on the number line.

This category includes:

• Integers, like -1, 0, and 1
• Fractions, such as \( \frac{1}{2} \) and 0.75
• Irrational numbers, like \( \sqrt{2} \) and \( \pi \)

When we talk about the domain of a function being all real numbers, we generally mean every conceivable number along the number line. However, in the context of rational functions, we sometimes exclude numbers that lead to undefined expressions (like division by zero). Therefore, in our function \( g(w) \), the real numbers do not include \( w = -4 \) and \( w = 4 \).
Real numbers thus provide a comprehensive framework for solving, graphing, and understanding functions across different mathematical contexts.