Problem 14 - College Algebra

Question

For the following exercises, find the domain of each function using interval notation. $$ f(x)=\frac{9}{x-6} $$

Step-by-Step Solution

Answer

The domain of the function is \((-\infty, 6) \cup (6, \infty)\).

Step 1: Identify the restriction

To determine the domain of a rational function, we need to find out when the denominator is equal to zero, as division by zero is undefined. For the function \( f(x) = \frac{9}{x-6} \), the restriction occurs when \( x - 6 = 0 \).

Step 2: Solve for the restricted value

Solve the equation \( x - 6 = 0 \) to determine the value of \( x \) that would make the denominator zero: \( x = 6 \). Therefore, \( f(x) \) is not defined at \( x = 6 \).

Step 3: Express the domain in interval notation

Since the function is undefined at \( x = 6 \), the domain includes all real numbers except 6. In interval notation, this is expressed as \((-\infty, 6) \cup (6, \infty)\).

Key Concepts

Rational Functions

A rational function is a type of function represented as the ratio of two polynomials. In mathematical terms, it looks like this: \[ f(x) = \frac{P(x)}{Q(x)} \] where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \) is not the zero polynomial. Rational functions are powerful tools in modeling many real-world situations because they can handle continuous data with exceptions where specific inputs do not result in valid outputs.

• They behave similarly to fractions, where central operations and properties are emphasized by the numerator and denominator.
• A critical aspect of rational functions is identifying where they are undefined, usually when the denominator equals zero.

This feature is essential for finding the domain of the function, which is the set of all possible input values \( x \). The undefined points become crucial restrictions to note when determining the domain.

Interval Notation

Interval notation is a concise way to describe a range of values that a function can accept as input or produce as output. It uses parentheses \(( )\) and brackets \([ ]\) to indicate open and closed intervals, respectively. Let's break this down further:

• Open Intervals: Denoted using parentheses, such as \((a, b)\), these indicate the range excludes the endpoints \(a\) and \(b\).
• Closed Intervals: Denoted using brackets, such as \([a, b]\), these signify that the range includes \(a\) and \(b\).
• Union: When multiple intervals combine to cover non-continuous ranges, a union symbol \(\cup\) shows this, like in \((a, b) \cup (c, d)\).

Using interval notation simplifies how we express domains, especially for rational functions where exceptions like division by zero need to be excluded. For example, with \(f(x) = \frac{9}{x-6}\), the domain excludes \(6\), so we express it as \((-\infty, 6) \cup (6, \infty)\). This means all real numbers except \(6\) are valid inputs.

Division by Zero

In mathematics, division by zero is an undefined operation, meaning there's no number you can multiply by zero to get a nonzero numerator. This concept becomes particularly important when dealing with rational functions.

• Why Undefined?: If we attempted to divide by zero, the operation wouldn't produce a finite number or meaningful result.
• Impact on Functions: In equations and functions, when the denominator of a rational function is zero, it indicates a point where the function doesn't exist, thereby directly affecting the domain.

For the function \(f(x) = \frac{9}{x-6}\), the expression becomes undefined if \(x = 6\), since it results in division by zero: \(\frac{9}{0}\). Understanding this restriction is critical when stating the domain of the function. It's these very points that determine what values need to be excluded from the set of possible inputs, ensuring that the function remains defined and sensible.