Problem 4 - College Algebra

Question

In Exercises \(1-8,\) find the domain of each rational function. $$g(x)=\frac{2 x^{2}}{(x-2)(x+6)}$$

Step-by-Step Solution

Answer

The domain of the function \( g(x)=\frac{2 x^{2}}{(x-2)(x+6)} \) is \( (-\infty, -6) \cup (-6, 2) \cup (2, +\infty) \)

Step 1: Identify the Rational Function

The rational function here is \( g(x)=\frac{2 x^{2}}{(x-2)(x+6)} \).

Step 2: Set the Denominator Equals to Zero

To find the values of x that are not in the domain, set the denominator equal to zero and solve for x.\[(x-2)(x+6) = 0\]

Step 3: Solve for 'x'

Solving the equation gives two values for \(x\):\n1. If \(x-2 = 0\), then \(x = 2\).\n2. If \(x+6 = 0\), then \(x = -6\).

Step 4: State the Domain

The domain of the function is all real numbers except for 2 and -6. This can be written as: \n\(x\in R, x \neq 2, x \neq -6\) in interval notation, it is \( (-\infty, -6) \cup (-6, 2) \cup (2, +\infty) \)

Key Concepts

Rational Functions

Rational functions are a fundamental type of function in algebra. They are defined as a ratio of two polynomials. This means you have a polynomial in the numerator and another polynomial in the denominator. For example, the function given in the exercise, \( g(x) = \frac{2x^2}{(x-2)(x+6)} \), is a rational function. The polynomial \(2x^2\) is in the numerator, and \((x-2)(x+6)\) is in the denominator.

It is crucial to understand that rational functions are only defined where their denominators are not zero. Wherever the denominator is zero, the function is undefined because division by zero is not possible. This concept is important when determining the function's domain, which specifies all possible input values (\(x\)) for which the function produces a valid output.

Denominator

A key feature of a rational function is its denominator. The denominator determines where the function can and cannot be evaluated. Understanding this part of the rational function is critical because it affects the domain directly.

In the example \( g(x) = \frac{2x^2}{(x-2)(x+6)} \), the denominator is \((x-2)(x+6)\). When finding the domain, it is essential to set the denominator equal to zero, because these are the points that are excluded from the domain. Solving \((x-2)(x+6) = 0\) allows us to find the exact values of \(x\) that make the function undefined. This step involves basic algebraic solutions that are vital when working with rational functions.

Domain Exclusion

Domain exclusion is an important concept when dealing with rational functions. It highlights the values of \(x\) that should be removed from the domain of the function. These exclusions occur precisely where the denominator equals zero.

For the function \( g(x) = \frac{2x^2}{(x-2)(x+6)} \), setting the denominator to zero gives \( (x-2)(x+6) = 0 \). Solving this equation provides the excluded values of \(x = 2\) and \(x = -6\). These are the values which make the denominator and thus the entire function undefined. It's critical to identify and exclude these from the function's domain to ensure it is properly defined.

Interval Notation

Interval notation is a concise way to describe the set of all numbers that comprise a function's domain. It is particularly useful for rational functions because it clearly displays which values are included or excluded.

For the function \( g(x) = \frac{2x^2}{(x-2)(x+6)} \), we determine that the domain excludes \(x = 2\) and \(x = -6\). Hence, the domain can be expressed in interval notation as:

• \( (-\infty, -6) \)
• \( \cup (-6, 2) \)
• \( \cup (2, +\infty) \)

This notation succinctly indicates that all real numbers are included except for the intervals where \(x = -6\) and \(x = 2\). It's an efficient way to visualize and communicate the acceptable inputs to rational functions.