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Chapter 6: Problem 92

Find the domain of \(f\). \(f(x)=\frac{3 x}{x^{2}-1}\)

Short Answer

Expert verified
The domain is \( (-\infty, -1) \cup (-1, 1) \cup (1, \infty) \).

Step by step solution

01

Identify the function type

The given function is a rational function of the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) = 3x \) and \( Q(x) = x^{2} - 1 \).
02

Determine where the denominator is zero

For rational functions, the function is undefined where the denominator is equal to zero. Set the denominator equal to zero: \( x^{2} - 1 = 0 \).
03

Solve for x

Solve the equation \( x^{2} - 1 = 0 \): \( x^{2} = 1 \) leads to \( x = \pm 1 \).
04

Express the domain

The domain includes all real numbers except those that make the denominator zero. Therefore, the domain is all real numbers except \( x = 1 \) and \( x = -1 \).
05

Write the domain in interval notation

Express the domain in interval notation: \( (-\infty, -1) \cup (-1, 1) \cup (1, \infty) \).

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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 expressions that represent the ratio of two polynomials. Generally, a rational function can be written as \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) \) is not the zero polynomial. These functions can model real-world situations, including rates of change and proportions. Key features of rational functions include their asymptotes and intervals of increase or decrease. Understanding these features helps to analyze the function's behavior and potential limitations or restrictions.
Undefined Values
Rational functions can be undefined at certain points. These points occur where the denominator, \( Q(x) \), equals zero because division by zero is undefined in mathematics. To find these undefined values for a given rational function, set the denominator equal to zero and solve for the variable. In the function \( f(x) = \frac{3x}{x^2 - 1} \), the denominator is \( x^2 - 1 \). Setting this equal to zero gives \( x^2 - 1 = 0 \), solving which results in \( x = \pm 1 \). Thus, the function is undefined at \( x = 1 \) and \( x = -1 \).
Interval Notation
Interval notation is a way of expressing the domain or range of a function without listing every individual number. Instead, it uses brackets and parentheses to denote intervals of numbers. For example, the interval \( (-鈭�, -1) \) includes all real numbers less than -1, while \( (1, 鈭�) \) includes all numbers greater than 1. To express the domain of a function in interval notation, determine the intervals over which the function is defined. For \( f(x) = \frac{3x}{x^2 - 1} \), the function is defined for all real numbers except where \( x = -1 \) or \( x = 1 \). Thus, the domain in interval notation is written as \( (-鈭�, -1) \cup (-1, 1) \cup (1, 鈭�) \).

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