Problem 7 - College Algebra

Question

For the following exercises, find the domain of the rational functions. $$ f(x)=\frac{x+1}{x^{2}-1} $$

Step-by-Step Solution

Answer

The domain of \(f(x) = \frac{x+1}{x^2-1}\) is \((-\infty, -1) \cup (-1, 1) \cup (1, \infty)\)."

Step 1: Identify the Denominator

For rational functions, the domain is all real numbers except those that make the denominator zero. The denominator of the function is \(x^2 - 1\).

Step 2: Set the Denominator Equal to Zero

To find which values are excluded from the domain, set the denominator equal to zero: \(x^2 - 1 = 0\).

Step 3: Solve for x

Solve the equation \(x^2 - 1 = 0\):\[x^2 = 1\]Then take the square root of both sides to find \(x\):\[x = \pm 1\]This means \(x = 1\) and \(x = -1\) are the values that make the denominator zero.

Step 4: State the Domain

The domain of the function excludes \(x = 1\) and \(x = -1\). Thus, the domain of \(f(x) = \frac{x+1}{x^2-1}\) is all real numbers except \(x = 1\) and \(x = -1\), which can be written in interval notation as \((-\infty, -1) \cup (-1, 1) \cup (1, \infty)\).

Key Concepts

Rational Functions

A rational function is simply a fraction where both the numerator and the denominator are polynomials. These functions often look like this: \( f(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomial expressions. The main thing to remember about rational functions is that the denominator \( q(x) \) can never be zero, because division by zero is undefined. This is crucial in determining the function's domain, which we'll discuss shortly. Rational functions can model a variety of real-life situations, but the requirement to avoid division by zero remains a constant consideration.

Denominator Equation

The denominator in a rational function is the key to finding out which values are not allowed in the function's domain. In the given exercise, the denominator is \( x^2 - 1 \). To find the values that make this denominator zero, you need to set up an equation: \( x^2 - 1 = 0 \). By solving this equation, you determine the points where the rational function is undefined.

• Set the equation equal to zero to determine problematic values.
• Think of the denominator as a guide to identifying excluded values in the real number set.

Once you've identified these values, you'll know which numbers can't be used in the function, helping to establish the domain accurately.

Real Numbers

In mathematics, real numbers include all the numbers you might think of on the number line: positive numbers, negative numbers, zero, and rational numbers like fractions and decimals. A rational function's domain typically includes all real numbers except those that make the denominator zero. When figuring out the domain of \( f(x) = \frac{x+1}{x^2-1} \), you have to exclude any 'real' values of \( x \) that make the denominator zero, specifically \( x = 1 \) and \( x = -1 \). It is important to be comfortable identifying such values within the continuum of real numbers as the first step in working with rational functions effectively.

Interval Notation

Interval notation is a way of expressing domains of functions succinctly by using intervals instead of long sentences or words. For instance, the domain of the function \( f(x) = \frac{x+1}{x^2-1} \) is all real numbers except \( x = 1 \) and \( x = -1 \). Using interval notation, this domain is expressed as \( (-\infty, -1) \cup (-1, 1) \cup (1, \infty) \).

• Use round brackets \(()\) for exclusive bounds where a particular value is not included, such as \((-\infty, -1)\).
• The symbol \(\cup\) indicates the union of separate intervals, meaning all numbers in the intervals are part of the domain except the excluded values.

This convenient notation is extremely helpful for clearly defining the domain of rational functions or any other mathematical expressions.