Problem 14 - College Algebra with Corequisite Support

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: Understand the Function

The function given is \(f(x) = \frac{9}{x-6}\). This is a rational function, where the numerator is 9 and the denominator is \(x-6\). Rational functions are undefined where their denominators are zero.

Step 2: Identify Situation for Undefined Values

For the function \(f(x) = \frac{9}{x-6}\), identify where the denominator becomes zero. \(x-6=0\) when \(x=6\). Thus, the function is undefined at \(x=6\).

Step 3: State the Domain in Interval Notation

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

Key Concepts

Rational Functions

Rational functions are expressions formed by the division of two polynomials. Generally, they are written in the form \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomial functions and \(Q(x) eq 0\). These functions can exhibit interesting behaviors, particularly where the denominator, \(Q(x)\), equals zero. In such cases, the function becomes undefined, leading to gaps or asymptotes in its graph.

Rational functions are widely studied in algebra due to their applications in modeling natural phenomena and physics, where rates of change and proportional relationships are involved. When you encounter such a function, it's essential to find where the function is undefined to understand its domain and behavior better. Always remember, the key with rational functions is to focus on the denominator and identify where it becomes zero.

Interval Notation

Interval notation is a concise way to describe a set of real numbers. It uses parentheses \(()\) and brackets \([]\) to indicate which endpoints are included in the set. Here's a quick breakdown:

• Parentheses \(()\): Used when an endpoint is not included in the interval. This often accompanies infinity, as infinity is not a number we can reach or include.
• Brackets \([]\): Indicate that an endpoint is included in the interval.

For example, the interval \((-fty, 6) \cup (6, fty)\) tells us that all numbers less than 6 and greater than 6 are included. However, the number 6 itself is not in the domain. Intervals can be unioned, represented by the symbol \(\cup\), to combine separate parts of a domain. This notation is especially useful in describing complex sets of numbers, like those encountered in rational functions, which may have one or more excluded values.

Undefined Values in Functions

Undefined values occur in functions where an operation doesn't have a valid result. In rational functions, this typically happens when you attempt to divide by zero, causing the function to be undefined at certain points. For the function \(f(x) = \frac{9}{x-6}\), for example, the value of \(x\) that makes the denominator zero is \(x=6\), hence the function is undefined at \(x=6\).

To find where a function is undefined, set the denominator equal to zero and solve for \(x\). It's crucial to identify these points because they affect the function's domain, limiting the real numbers you can use as inputs. Once an undefined value is identified, it is removed from the domain, often leaving a gap or vertical asymptote in the graph of the function. Recognizing undefined values is key in accurately determining how a function behaves across its entire domain.