Problem 49 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Domain: All real numbers except \( x = 1 \) and \( x = -1 \).

Step 1: Identify the Problematic Denominator

To find the domain of the function \( f(x) = \frac{x+2}{x^{2}-1} \), start by identifying when the denominator equals zero because a function is undefined when the denominator is zero.

Step 2: Set the Denominator Equal to Zero

Set the denominator \( x^{2} - 1 \) equal to zero and solve for \( x \). This gives \( x^{2} - 1 = 0 \).

Step 3: Solve the Equation for Excluded Values

Solve the equation \( x^{2} - 1 = 0 \). Factor it as \( (x - 1)(x + 1) = 0 \), so \( x = 1 \) and \( x = -1 \). These are values where the function is undefined.

Step 4: Write the Domain Excluding Problematic Values

The domain of \( f(x) \) is all real numbers except where the denominator is zero. Thus, the domain is all real numbers \( \mathbb{R} \) except \( x = 1 \) and \( x = -1 \).

Key Concepts

Rational Functions

Rational functions are a type of function expressed as the ratio of two polynomials. In simpler terms, think of them as fractions where one polynomial is divided by another. The general form of a rational function is given by:

• The numerator is a polynomial, represented by the function at the top of the fraction.
• The denominator is another polynomial, which forms the bottom part of the fraction.

In the rational function we are examining, which is defined as \( f(x) = \frac{x+2}{x^2 - 1} \), "\( x + 2 \)" is the numerator and "\( x^2 - 1 \)" is the denominator. Understanding these components is essential, as they each play a role in the behavior and characteristics of the function. Rational functions are undefined when the denominator is zero, making it crucial to determine these values to understand the function's domain.

Denominator

The denominator is the polynomial located below the line in a rational function. It provides the divisor that determines the values that are potentially problematic in the function. For our function \( f(x) = \frac{x+2}{x^2 - 1} \), the denominator is \( x^2 - 1 \). When working with rational functions, finding when the denominator equals zero helps identify the points where the function is undefined. To do so, we set the denominator equal to zero: \[ x^2 - 1 = 0 \]Solving this equation reveals the values of \( x \) that cause the denominator to zero out, hence making the rational function undefined. These solutions need to be excluded from the function's domain, as including them would result in division by zero, which is mathematically undefined.

Excluded Values

Excluded values in a rational function refer to specific inputs where the function does not have a defined output. These are usually the values that make the denominator zero. To determine these, the denominator is set to zero and solved for \( x \). Take our example, \( x^2 - 1 = 0 \). This equation factors into:

• \( (x - 1)(x + 1) = 0 \)

Solving this gives the solutions \( x = 1 \) and \( x = -1 \). These are the excluded values for our function \( f(x) = \frac{x + 2}{x^2 - 1} \). These particular values cause division by zero, rendering the function undefined at these points. Therefore, they are excluded when defining the domain.

Real Numbers

Real numbers encompass all numbers that can be found on the number line. This includes both rational numbers (like fractions and whole numbers) and irrational numbers (which cannot be expressed as a simple fraction). The domain of many functions is often represented using real numbers, since they provide a complete set of values that a function can accept as inputs.For the function \( f(x) = \frac{x+2}{x^2 - 1} \), its domain consists of all real numbers, except those that make the denominator zero. So, the domain excludes \( x = 1 \) and \( x = -1 \). In set notation, this can be expressed as:\[ \{ x \in \mathbb{R} \mid x eq 1, x eq -1 \} \]Understanding real numbers is key to grasping domains of functions, as it allows us to easily see the range of possible inputs and identify those that need to be excluded for specific functions.