Problem 18 - College Algebra

Question

For the following exercises, find the domain of each function using interval notation. $$ f(x)=\frac{1}{x^{2}-x-6} $$

Step-by-Step Solution

Answer

The domain is \((-\infty, -2) \cup (-2, 3) \cup (3, \infty)\)."

Step 1: Understand the Function

The function given is \( f(x) = \frac{1}{x^2 - x - 6} \). This is a rational function, which means it has a numerator and a denominator. The domain of the function is all real numbers except those which make the denominator zero, as these result in undefined values.

Step 2: Set the Denominator Equal to Zero

To find which values of \( x \) make the function undefined, set the denominator equal to zero: \( x^2 - x - 6 = 0 \). This will help us find the values of \( x \) to exclude from the domain.

Step 3: Solve the Quadratic Equation

Solve the equation \( x^2 - x - 6 = 0 \) using factoring:1. Factor the quadratic: \((x - 3)(x + 2) = 0\).2. Set each factor equal to zero: \(x - 3 = 0\) or \(x + 2 = 0\).3. Solve for \( x \): \(x = 3\) or \(x = -2\). These are the values that make the denominator zero.

Step 4: Write the Domain in Interval Notation

With \( x = 3 \) and \( x = -2 \) excluded from the domain, the domain in interval notation is all real numbers except \( -2 \) and \( 3 \). Therefore, the domain is \(( -\infty, -2 ) \cup (-2, 3) \cup (3, \infty)\).

Key Concepts

Rational Functions

Rational functions form a central part of algebra and calculus. A rational function is defined as the ratio of two polynomials. Think of it like a fraction, where you have a numerator and a denominator, but with polynomial expressions. This gives them their distinct nature and interesting properties.
Understanding rational functions is essential because they can model various real-world situations.

• The behavior of these functions relies heavily on their denominators. Specifically, wherever the denominator is zero, the function becomes undefined.
• This undefined nature means we must carefully analyze rational functions to determine their domain correctly.

In the example provided, the rational function was expressed as \( f(x) = \frac{1}{x^2 - x - 6} \). The focus is on finding where its denominator equals zero, as these points will need to be excluded from the function's domain.

Interval Notation

Interval notation is an efficient way of representing a set of numbers, typically used to describe domains or ranges of functions. It helps to express large sets compactly by showing the start and end of an interval.
When using interval notation, you need to remember a few key points:

• Round brackets \(( )\) indicate that a number is not included in the interval. This is also known as an "open" interval.
• Square brackets \([ ]\) mean that a number is included, representing a "closed" interval.
• The union symbol \( \cup \) is used to combine multiple intervals into one complete set.

In our example, the intervals \(( -\infty, -2 ) \), \(( -2, 3 )\), and \(( 3, \infty) \) were used. The round brackets tell us that the values \(-2\) and \(3\) themselves are not included in the domain. The union symbol combines these separate intervals into the full domain of the rational function.

Undefined Values

In functions, undefined values play a crucial role in determining the domain. For rational functions like the one in our example, undefined values occur when the denominator equals zero. This is because division by zero is something we cannot do in mathematics, as it does not result in a valid or meaningful number.
To manage undefined values:

• Identify the parts of your function where it becomes undefined. This typically means setting the denominator to zero and solving for the variable.
• Exclude these solutions from the function's domain to avoid undefined expressions.

The particular rational function \( f(x) = \frac{1}{x^2 - x - 6} \) shows this, where \( x = 3 \) and \( x = -2 \) are values that make the denominator zero. Consequently, these values are excluded from the domain, ensuring the function remains defined for all other real numbers.

Quadratic Equations

Quadratic equations appear frequently in the study of algebra. They are polynomials of degree 2, typically in the form \( ax^2 + bx + c = 0 \). Solving these equations helps find the roots, or the "solutions," where the equation is equal to zero.
There are different methods for solving quadratic equations, such as:

• Factoring the quadratic, if possible, into two binomial expressions.
• Using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Completing the square, to transform the equation into a perfect square trinomial.

In our example, factoring was used to find values of \( x \) that would make the denominator zero: \( (x - 3)(x + 2) = 0 \). This results in the roots \( x = 3 \) and \( x = -2 \), which are critical for understanding the function's undefined points and subsequently, its domain.