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Chapter 1: Problem 12

Find the domain of each function $$ f(x)=\frac{6}{x-8} $$

Short Answer

Expert verified
The domain is all real numbers except \( x = 8 \); in interval notation: \((-\infty, 8) \cup (8, \infty)\).

Step by step solution

01

Identify Potential Denominators that Cause Division by Zero

To find the domain of the function \( f(x) = \frac{6}{x-8} \), we need to determine where the function is undefined. A rational function is undefined when its denominator equals zero. Here, the denominator is \( x - 8 \).
02

Solve the Equation

Set the denominator equal to zero to find any restrictions: \[ x - 8 = 0 \]Solving for \( x \), we add 8 to both sides to get:\[ x = 8 \]
03

Determine the Domain Excluding Restrictions

The value \( x = 8 \) makes the denominator zero, so it is not included in the domain of the function. Therefore, the domain of \( f(x) \) is all real numbers except \( x = 8 \). In interval notation, this is represented as:\[ (-\infty, 8) \cup (8, \infty) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Functions
Rational functions are a type of function that are formed by the division of two polynomials. Specifically, a rational function can be expressed in the general form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \).

Key characteristics of rational functions include:
  • They can represent fractions where the numerator and denominator are polynomials.
  • They could potentially be undefined at certain points because division by zero is not allowed.
  • The graphs of rational functions can have vertical asymptotes, horizontal asymptotes, and holes, depending on the specific equations being graphed.


Importantly, understanding the behavior of rational functions is crucial in calculus and other branches of mathematics because they model many real-world scenarios.
Division by Zero
One of the most critical aspects of dealing with rational functions is handling division by zero. When we say "division by zero," we refer to a situation where the denominator of a fraction becomes zero. Mathematically, division by zero is undefined, which means it does not produce a valid result. Consequently, we must be careful to exclude any values from the domain that would make the denominator zero.

In calculating the domain of a function like \( f(x) = \frac{6}{x - 8} \), we start by setting the denominator equal to zero:
  • \( x - 8 = 0 \)
  • This equation reveals that \( x = 8 \) will make the denominator zero, thus the function is undefined at this point.

Therefore, to avoid division by zero, we should exclude \( x = 8 \) from the domain of the function.
Interval Notation
In mathematics, interval notation is a concise way of expressing subsets of real numbers, particularly when describing the domain of a function. It consists of using parentheses \(( )\) or brackets \([ ]\) to depict intervals.

Here's a quick guide to understanding interval notation:
  • Parentheses \(( )\) are used when an endpoint is not included in the interval, indicating it extends up to but does not include the boundary number.
  • Brackets \([ ]\) are used when an endpoint is included.
  • A union operator \(\cup\) combines two distinct intervals into a single set, commonly used when a domain skips over a singular value.

For the rational function \( f(x) = \frac{6}{x-8} \), the domain excludes \( x=8 \), so the domain in interval notation is represented as \((-\infty, 8) \cup (8, \infty)\). This notation captures all possible real numbers except the forbidden value, \( x = 8 \).

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Most popular questions from this chapter

For each table below, select whether the table represents a function that is increasing or decreasing, and whether the function is concave up or concave down. $$ \begin{array}{|l|l|} \hline x & g(x) \\ \hline 1 & -200 \\ \hline 2 & -190 \\ \hline 3 & -160 \\ \hline 4 & -100 \\ \hline 5 & 0 \\ \hline \end{array} $$

Let \(g(p)=6-2 p\) a. Evaluate \(g(0)\) b. Solve \(g(p)=0\)

Sketch a reasonable graph for each of the following functions. [UW] a. Distance of your big toe from the ground as you ride your bike for 10 seconds. b. Your height above the water level in a swimming pool after you dive off the high board. c. The percentage of dates and names you'll remember for a history test, depending on the time you study.

Starting with the graph of \(f(x)=4^{x}\) write the equation of the graph that results from a. reflecting \(f(x)\) about the \(x\) -axis b. reflecting \(f(x)\) about the \(y\) -axis, shifting right 4 units, and up 2 units

Determine the interval(s) on which the function is increasing and decreasing. $$ f(x)=4(x+1)^{2}-5 $$
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