Problem 30 - Elementary and Intermediate Algebra

Question

Find the domain of each rational function. Express your answer in words and using interval notation. See Example 2. $$f(x)=\frac{x^{2}+36}{x^{2}-36}$$

Step-by-Step Solution

Answer

The domain is all real numbers except \( x = 6 \) and \( x = -6 \), expressed as \( (-\infty, -6) \cup (-6, 6) \cup (6, \infty) \).

Step 1: Understand the Rational Function

A rational function is given by \( f(x) = \frac{x^{2} + 36}{x^{2} - 36} \). The domain of a rational function includes all real numbers except where the denominator is equal to zero.

Step 2: Set the Denominator Equal to Zero

We need to find the values of \( x \) that make the denominator zero, so set \( x^{2} - 36 = 0 \).

Step 3: Solve the Equation for x

Solve the equation \( x^2 - 36 = 0 \) by adding 36 to both sides, giving \( x^2 = 36 \). Taking the square root of both sides results in \( x = 6 \) and \( x = -6 \).

Step 4: Exclude Critical Points from the Domain

The critical points, \( x = 6 \) and \( x = -6 \), are the values that make the denominator zero, thus are not part of the domain.

Step 5: Express the Domain

The domain of \( f(x) \) is all real numbers except \( x = 6 \) and \( x = -6 \). In interval notation, this is expressed as \( (-\infty, -6) \cup (-6, 6) \cup (6, \infty) \).

Key Concepts

Understanding the Domain of a Function

The domain of a function refers to the set of all possible input values (usually \( x \,\) values) for which the function is defined. In simpler terms, it tells us which values we can plug into the function without running into any mathematical problems, such as division by zero. For rational functions, which are fractions with polynomials in the numerator and the denominator, the primary concern is ensuring that the denominator does not equal zero. This is because division by zero is undefined in mathematics and causes the function to "break" at those points.
To identify the domain of a rational function, follow these steps:

• Focus on the denominator of the fraction.
• Set the denominator equal to zero.
• Solve the equation to find the critical points (the values that make the denominator zero).
• Exclude these critical points from the set of all real numbers to find the domain.

By excluding problematic values (where the denominator is zero), you establish the domain where the function behaves correctly.

Working with Interval Notation

Interval notation is a way of writing subsets of the real number line. It's a helpful tool to express the domain of a function, particularly when the domain includes entire ranges of numbers rather than discrete points. Interval notation uses parentheses \( ( ) \,\) and brackets \( [ ] \,\) to indicate which numbers are included or excluded in these ranges.

Here is how interval notation works:

• Parentheses \( ( ) \,\) are used when a number is not included in the interval, which is known as an "open" interval.
• Brackets \( [ ] \,\) are used when a number is included, which creates a "closed" interval.

For example, the interval \( (-\infty, -6) \cup (-6, 6) \cup (6, \infty) \,\) shows three parts:

• The first part \( (-\infty, -6) \,\) means every real number less than -6 is included.
• The second part \( (-6, 6) \,\) covers numbers between -6 and 6, not including -6 or 6 themselves.
• And \( (6, \infty) \,\) refers to numbers greater than 6, but not including 6.

This type of notation is concise and clear, providing an efficient way to describe infinite ranges.

The Role of the Denominator in Rational Functions

The denominator of a rational function is crucial in determining the function's domain. In the realm of rational functions, the denominator is what dictates the function's well-defined nature. It consists of polynomial expressions which, if evaluated to zero, can lead to undefined values for the function.

Consider the function \( f(x) = \frac{x^{2} + 36}{x^{2} - 36} \,\). Here, our denominator is \( x^{2} - 36 \,\). If we set this equal to zero, we get \( x^{2} - 36 = 0 \,\). Solving this gives us \( x = 6 \,\) and \( x = -6 \,\), which are the numbers that make the denominator zero.
When evaluating the denominator's impact:

• Find solutions to the equation where the denominator equals zero.
• These solutions are the critical points that need to be excluded from the domain.
• Once identified, express the domain excluding these numbers.

In this example, excluding \( x = 6 \,\) and \( x = -6 \,\) means \( f(x) \,\) is undefined at those points, so they are left out of the domain. This analysis helps keep the function valid and can effectively define the domain, ensuring mathematical consistency.