Problem 4 - Algebra and Trigonometry

Question

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 \((-∞,-6) ∪ (-6,2) ∪ (2,∞)\).

Step 1: Identify Denominator

The denominator of the function is \((x-2)(x+6)\). Set this equal to zero to find the \(x\)-values that we should exclude from the domain.

Step 2: Solve For Excluded Values

Solving the equations \(x-2 = 0\) and \(x+6 = 0\) gives \(x=2\) and \(x=-6\) respectively.

Step 3: Formulate the Domain

The function will be undefined for \(x=2\) and \(x=-6\), therefore these values are not in the domain. All other real numbers are in the domain of \(g(x)\). So the domain of \(g(x)\) is all real numbers except \(-6\) and \(2\). We can represent this in interval notation as \((-∞,-6) ∪ (-6,2) ∪ (2,∞)\).

Key Concepts

Rational Function

Rational functions are expressions of the form \(\frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomials and \(q(x)\) is not the zero polynomial. These functions can be thought of as ratios, similar to how fractions are defined with a numerator and a denominator. Because the denominator defines where the function can go undefined, it's crucial to determine those points where the function might not work properly. That's why we need to be particularly focused on the zeros of the denominator. It's important to express these rules and ideas clearly:

• The numerator \(p(x)\) dictates the zeros of the function.
• The denominator \(q(x)\) guides us to the values where the function is undefined.

Understanding the structure of a rational function helps us determine its characteristics, like the domain. By identifying what makes the denominator zero, we find those specific \(x\) values that must be excluded from the domain.

Excluded Values

Excluded values in rational functions are the specific \(x\) inputs that make the function undefined. To find these, we look at the denominator of the rational expression and set it equal to zero, because division by zero is undefined in mathematics. This process helps determine which values should be left out from the function's domain.

For example, given the function \(g(x)=\frac{2x^2}{(x-2)(x+6)}\), finding where the denominator \((x-2)(x+6)=0\) gives us the excluded values \(x=2\) and \(x=-6\). This means:

• If \(x\) is \(2\), then \((x-2)\) makes the denominator zero.
• If \(x\) is \(-6\), then \((x+6)\) makes the denominator zero.

Leaving these values out when defining the domain ensures the function remains operational. Hence, it's crucial to always check the denominator for zeros.

Interval Notation

Interval notation is a method for describing sets of numbers. It is particularly useful when defining domains of functions such as rational functions. This notation uses parentheses \(()\) and brackets \([]\) to describe open or closed intervals, respectively.

In the context of our function \(g(x)=\frac{2x^2}{(x-2)(x+6)}\), we found that the excluded values are \(x=2\) and \(x=-6\). Therefore, the domain includes all real numbers except these two.

• The interval \((-∞,-6)\) represents all numbers less than \(-6\).
• \((-6,2)\) includes numbers between \(-6\) and \(2\) but does not include \(-6\) or \(2\).
• \((2,∞)\) covers all the numbers greater than \(2\).

Using interval notation, the domain of the function is written as:
\((-∞,-6) ∪ (-6,2) ∪ (2,∞)\).

This notation provides a clear and concise way to specify which \(x\) values keep the function defined and are part of its domain.