Problem 43 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The domain is all real numbers except \( x = 1 \) and \( x = -1 \).

Step 1: Understand the Function

The function provided is a rational function \( f(x) = \frac{x+2}{x^2-1} \). Rational functions can be undefined where their denominators are equal to zero, so we must determine when the denominator becomes zero to find its domain.

Step 2: Set the Denominator Equal to Zero

To find where the function is undefined, identify when the denominator \( x^2 - 1 \) equals zero: \[ x^2 - 1 = 0 \] .

Step 3: Solve for x

Solve the equation \( x^2 - 1 = 0 \). Add 1 to both sides to get \( x^2 = 1 \). Then, take the square root of both sides to find \( x = \pm 1 \). Thus, the function is undefined at \( x = 1 \) and \( x = -1 \).

Step 4: Determine the Domain

Since the function is undefined at \( x = 1 \) and \( x = -1 \), these values must be excluded from the domain. Therefore, the domain of the function \( f(x) = \frac{x+2}{x^2-1} \) is all real numbers except \( x = 1 \) and \( x = -1 \). In interval notation, this is written as \( (-\infty, -1) \cup (-1, 1) \cup (1, \infty) \).

Key Concepts

Rational Functions

A rational function is essentially a fraction where both the numerator and the denominator are polynomials. In simpler terms, it's like a division problem within mathematics using two polynomial expressions. The function \( f(x) = \frac{x+2}{x^2-1} \) is an excellent example of a rational function. Here, \( x+2 \) is our numerator, while \( x^2-1 \) is the denominator. The overall behavior and properties of the function depend a lot on its denominator.

These functions are unique because they can behave quite differently from other types of functions, such as linear or quadratic functions. For instance, rational functions can have vertical asymptotes, which occur where the function becomes undefined and approaches infinity. The location of these asymptotes depends on the values that make the denominator zero.

• Numerator: The top part of the fraction, dictates the function's value when the denominator is non-zero.
• Denominator: The bottom part of a fraction, determines where the function could be undefined or have asymptotes.

Rational functions play a crucial role in calculus and real-world applications like physics and economics.

Undefined Points

A function is considered undefined at points where it cannot compute a value. For rational functions like \( f(x) = \frac{x+2}{x^2-1} \), undefined points arise when the denominator equals zero. These specific values are critical because they represent breaks or discontinuities in the function.

To find these undefined points in our example, we set the denominator \( x^2 - 1 \) equal to zero and solve for \( x \). This leads to the equation \( x^2 - 1 = 0 \). Solving gives \( x = \pm 1 \). Therefore, the function is undefined at \( x = 1 \) and \( x = -1 \).

Understanding undefined points helps us describe the domain of the function. In practical terms, these are values we avoid in calculations to prevent errors or infinite results.

Interval Notation

Interval notation is a concise way to describe a set of numbers, often used to specify the domain of a function. It effectively communicates which numbers are included or excluded along a number line. In the case of \( f(x) = \frac{x+2}{x^2-1} \), interval notation helps identify the range of \( x \) values where the function remains valid.

• Parentheses \(( )\): Indicate that an endpoint value is excluded from the interval.
• Brackets \([ ]\): Indicate that an endpoint value is included in the interval (though not used in this particular domain).

For this function, the domain excludes \( x = 1 \) and \( x = -1 \), leading to the expression: \((-\infty, -1) \cup (-1, 1) \cup (1, \infty)\).This expression means the function is valid for all real numbers except \( x = -1 \) and \( x = 1 \), highlighting the role of interval notation in set theory and calculus.

Denominator Zero

The denominator of a rational function is key to understanding when the function is undefined. It's also central to calculating the domain of the function. When a denominator equals zero, the function's value becomes undefined, creating potentially infinite values known as vertical asymptotes.

• Zero Denominator: Creates undefined points in the function leading to discontinuities.
• Vertical Asymptotes: Occur at these points, where the function rapidly increases or decreases without bounds.

For \( f(x) = \frac{x+2}{x^2-1} \), we examine the denominator \( x^2 - 1 \). Solving the equation \( x^2 - 1 = 0 \) gives \( x = \pm 1 \). The function cannot have these values in its domain, as substituting them would make the denominator zero, resulting in undefined behavior. Hence, understanding the importance of ensuring the denominator is not zero is fundamental in safely working with rational functions.