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Chapter 7: Problem 6

For each rational function, find all numbers that are not in the domain. Then give the domain, using set-builder notation. $$ f(x)=\frac{9 x+8}{x} $$

Short Answer

Expert verified
The domain of \( f(x) = \frac{9x + 8}{x} \) is \{ x \in \mathbb{R} \mid x \eq 0 \}\. \( x = 0 \) is not in the domain.

Step by step solution

01

- Identify the Rational Function

The given rational function is \( f(x) = \frac{9x + 8}{x} \). A rational function is the ratio of two polynomials.
02

- Determine Restrictions on the Denominator

For the function to be defined, the denominator cannot be zero. Identify the denominator of the function: it is \( x \). Set the denominator equal to zero and solve: \( x = 0 \). Therefore, \( x = 0 \) is not in the domain of the function.
03

- Express the Domain in Set-Builder Notation

Using set-builder notation, the domain consists of all real numbers except \( x = 0 \). This can be written as: \(\text{Domain} = \{ x \in \mathbb{R} \mid x \eq 0 \} \)\

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Key Concepts

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

headline of the respective core concept
The exercise involves finding the domain of a given rational function, specifically determining which values of x make the function undefined and expressing the valid domain using set-builder notation. To fully understand this, we need to dive deeper into a few key concepts. Let's break these down and make them easy to understand for everyone.
Rational Functions
A rational function is a fraction where both the numerator and the denominator are polynomials. Simply put, it's any function that can be written in the form \[ f(x) = \frac{P(x)}{Q(x)} \]where \(P(x)\) and \(Q(x)\) are polynomials.
Examples of rational functions include:
  • \( f(x) = \frac{1}{x} \)
  • \( g(x) = \frac{x^2 - 1}{x + 2} \)
  • \( h(x) = \frac{x + 4}{x^2 - x - 6} \)
Rational functions can exhibit interesting behaviors, like vertical asymptotes and holes in their graphs, due to restrictions in their domains.
Domain Restrictions
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, the most common restriction is that you cannot divide by zero. This means we need to find values of x that make the denominator equal to zero, because these values are not allowed.
Here's the step-by-step process to determine domain restrictions:
  • Identify the denominator of the rational function.
  • Set the denominator equal to zero and solve for x.
  • The solutions to the equation in step 2 are the x-values that need to be excluded from the domain.

For the given function \( f(x) = \frac{9x + 8}{x} \), the denominator is x. Setting it to zero, we get \( x = 0 \). Therefore, \( x = 0 \) is not in the domain.
Set-Builder Notation
Set-builder notation is a concise way to describe a set by specifying properties that its members must satisfy. Here鈥檚 how it works:
  • Use curly braces \( \{\} \).
  • Include a variable, typically \( x \), followed by a vertical bar or colon \( | \) which means 'such that.'
  • Specify the condition(s) that members of the set must satisfy.

For example, the domain of our function in set-builder notation is written as: \[ \{ x \in \mathbb{R} \mid x eq 0 \} \]
This reads as 'the set of all real numbers x such that x is not equal to 0.'
Using set-builder notation provides a clear and efficient way to communicate the domain restrictions of rational functions like the one in our exercise.

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Most popular questions from this chapter

Simplify. $$ \frac{\frac{5}{6}}{\frac{4}{9}} $$

Multiply or divide as indicated. $$ \frac{(x+3)(x-4)}{(x-4)(x+2)} \cdot \frac{(x+5)(x-6)}{(x+3)(x-6)} $$

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Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{3 y}{y^{2}+y z-2 z^{2}}+\frac{4 y-1}{y^{2}-z^{2}} $$

Concept Check Write each formula using the "language" of variation. For example, the formula for the circumference of a circle, \(C=2 \pi r,\) can be written as "The circumference of a circle varies directly as the length of its radius." \(S=4 \pi r^{2},\) where \(S\) is the surface area of a sphere with radius \(r\)
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