Problem 19 - College Algebra with Corequisite Support

Question

For the following exercises, find the domain of each function using interval notation. \(f(x)=\frac{2 x^{3}-250}{x^{2}-2 x-15}\)

Step-by-Step Solution

Answer

Domain: \((-\infty, -3) \cup (-3, 5) \cup (5, \infty)\).

Step 1: Identify Denominator Constraint

The domain of a function is restricted by values that make the denominator zero. So, let's set the denominator equal to zero and solve for \(x\): \(x^2 - 2x - 15 = 0\).

Step 2: Factor the Denominator

The quadratic can be factored as \((x-5)(x+3) = 0\).

Step 3: Solve for Restricted Values

Set each factor to zero to find the values of \(x\) that are not in the domain: \(x-5=0\) gives \(x=5\), and \(x+3=0\) gives \(x=-3\).

Step 4: Express the Domain in Interval Notation

Since these two values make the denominator zero, they are excluded from the domain. The domain in interval notation is \((-\infty, -3) \cup (-3, 5) \cup (5, \infty)\).

Key Concepts

Interval notation

Interval notation is a way to describe the domain of a function. It is compact and visually clear, making it easier to understand which values are included or excluded from a function's domain. In interval notation, we use parentheses

• ( ) to denote intervals that do not include the endpoints, also known as open intervals.
• [ ] to show closed intervals that include the endpoints.

For example, the interval

• (- abla, -3) means all real numbers less than -3.
• (-3, 5) consists of all numbers between -3 and 5, but not including -3 and 5 themselves.
• (5, abla) means all numbers greater than 5.

These intervals can be combined using the union symbol ∪, which indicates that any of the intervals could contain the variable value. This notation clearly shows where a function is defined and how exclusions affect it, guiding the understanding of a function's behavior across its domain.

Factoring quadratic equations

Factoring quadratic equations is a method used to simplify expressions and solve for variables. A quadratic equation usually takes the form \(ax^2 + bx + c = 0\). Finding factors helps break it into simpler components, often making it easier to identify constraints in a function's domain.
For example, consider the quadratic expression \(x^2 - 2x - 15 = 0\). Using factoring, we can rewrite it as

• \((x - 5)(x + 3) = 0\),

enabling us to identify the solutions \(x = 5\) and \(x = -3\). These values are critical as they represent points where the original equation equates to zero.
Factoring not only helps in identifying roots but also plays a crucial role in determining the domain of a function, especially when connected to denominator constraints.

Denominator constraints

Denominator constraints are critical when determining the domain of a function, particularly involving fractions. When we have a function of the form \(f(x) = \frac{P(x)}{Q(x)}\), the denominator \(Q(x)\) plays a crucial role. It's vital to identify where \(Q(x) = 0\) because these values cause the function to become undefined.
In our problem, the denominator is

• \(x^2 - 2x - 15\)

and setting it equal to zero helps pinpoint values that are not part of the function's domain.
These solutions, derived from the quadratic factoring, reveal where the function's denominator vanishes, thus restricting the domain due to division by zero being undefined in mathematics. Understanding denominator constraints ensures a clear definition of where functions hold validity.

Excluded values in domain

Excluded values in the domain of a function are those for which the function does not exist or becomes undefined. In the context of a rational function, this often occurs where the denominator becomes zero.
When we factorized the denominator

• \((x - 5)(x + 3)\)

we identified that \(x = 5\) and \(x = -3\). These are the values that were excluded from our domain. Therefore, the function \(f(x)=\frac{2x^{3}-250}{x^{2}-2x-15}\) does not exist when \(x\) is 5 or -3.
This understanding is foundational to sketching function graphs and analyzing their properties. This is why excluded values are crucial, indicating discontinuities or undefined behaviors in mathematical functions. They help plot more accurate representations and predictions of a function's behavior across its entire domain.