Problem 10 - College Algebra and Calculus: An Applied Approach

Question

Find the domain of the expression.\(\frac{4 x+3}{x^{2}-36}\)

Step-by-Step Solution

Answer

The domain of the function is all real numbers except -6 and 6.

Step 1: Identify the Denominator

The denominator of the function is \(x^{2}-36\).

Step 2: Set the Denominator Equal to Zero

In order to find the values that are not in the domain of the function, set the denominator equal to zero. This gives us the equation \(x^{2}-36=0\).

Step 3: Solve for x

When we solve the equation \(x^{2}-36=0\), we get \(x = 6\) and \(x = -6\). These are the values that are not in the domain of the function because they make the denominator zero and the function undefined.

Step 4: Write the Domain of the Function

The domain of the function is all real numbers except -6 and 6.

Key Concepts

Solving Quadratic Equations

In algebra, solving quadratic equations is one of the foundational skills necessary to advance in the subject. A quadratic equation can be easily recognized by its general form: ewline ewline \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. The solutions to these equations are known as the roots or zeros of the quadratic function. ewline ewline There are various methods to solve quadratic equations, such as factoring, using the quadratic formula, completing the square, or graphing. The most suitable method often depends on the specific form of the quadratic equation in question. ewline ewline Let's consider the equation \( x^2 - 36 = 0 \). This is a quadratic equation that can be solved by factoring: ewline ewline \( (x - 6)(x + 6) = 0 \). ewline ewline Setting each factor equal to zero gives us the values of \( x \) that solve the equation: ewline ewline \( x - 6 = 0 \) or \( x + 6 = 0 \), which yield \( x = 6 \) and \( x = -6 \), respectively. Understanding how to identify and solve these equations is critical when determining the function domain exclusion, another fundamental concept in algebra.

Function Domain Exclusion

The domain of a function consists of all the input values (usually represented by \( x \)) for which the function is defined. In other words, for any input value within the domain, the function must produce a valid output. However, certain values may need to be excluded from the domain to ensure the function's output is meaningful. ewline ewline

Identifying Exclusions through Quadratic Equations

In rational functions, which involve ratios of polynomials, domain exclusions typically arise when the denominator is equal to zero. As seen in our example, the denominator is a quadratic expression that can be factored into \( (x - 6)(x + 6) \). We exclude the values of \( x \) that make the denominator zero – in this case, \( x = 6 \) and \( x = -6 \). ewline ewline

Writing the Domain

After identifying these exclusions, we express the domain as all real numbers except for the excluded values: \( x \in \mathbb{R} \setminus \{-6, 6\} \). This ensures that the function doesn't produce undefined results. Knowing how to find these exclusions is vital for understanding the scope and limitations of a function.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. Understanding these expressions is essential for grasping the concept of domains in algebra. ewline ewline

Characteristics of Rational Expressions

A hallmark of rational expressions is that they can be undefined when their denominator is zero, as no number can be divided by zero. This characteristic leads to exclusions in the domain, as described previously. For example, in the rational function \( \frac{4x + 3}{x^2 - 36} \), the expression is undefined when \( x^2 - 36 = 0 \). ewline ewline

Simplifying and Manipulating

It's also common to simplify rational expressions by factoring and canceling out common factors from the numerator and the denominator. However, when dealing with simplification, it is critical not to lose sight of the excluded values that were identified before any simplification took place. These values remain excluded from the domain regardless of the simplification. ewline ewline Rational expressions are vital when working with functions, especially when solving equations or working with function operations such as addition, subtraction, multiplication, and division. Recognizing and dealing with domain exclusions is a key part of understanding and working with these algebraic structures.