Problem 21 - Intermediate Algebra

Question

Specify the domain for each of the functions. $$f(x)=\frac{-4}{x^{2}+6 x}$$

Step-by-Step Solution

Answer

Domain: \((-\infty, -6) \cup (-6, 0) \cup (0, \infty)\)

Step 1: Understand the Function Type

The function given is a rational function. Its domain is all real values of \(x\) except where the denominator is zero.

Step 2: Identify the Denominator

Identify the expression in the denominator of the function, which is \(x^2 + 6x\). The domain will exclude values that make this expression equal to zero.

Step 3: Set the Denominator Equal to Zero

Set the expression \(x^2 + 6x\) equal to zero and solve for \(x\) to find the values that are not in the domain: \( x^2 + 6x = 0 \).

Step 4: Solve the Equation

Factor the equation: \( x(x + 6) = 0 \). This yields solutions \(x = 0\) and \(x = -6\).

Step 5: Define the Domain

The domain of \(f(x)\) is all real numbers except \(x = 0\) and \(x = -6\). In interval notation, this is: \((-\infty, -6) \cup (-6, 0) \cup (0, \infty)\).

Key Concepts

Rational Functions

Rational functions are mathematical expressions formed by the ratio of two polynomials. The general form of a rational function is \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. They are pivotal in understanding various mathematical relationships because they describe a wide range of behaviors.

• A key characteristic of rational functions is that they can be undefined if the denominator \( Q(x) \) equals zero.
• This makes finding the domain crucial, as it tells us the set of all possible input values \( x \) for which the function is defined.

Rational functions can be used to model real-world scenarios, including rates and other comparative relationships.

Factoring Quadratics

Factoring quadratics is an essential skill when working with rational functions, especially for finding the domain. A quadratic is an expression of the form \( ax^2 + bx + c \). The process of factoring involves writing it as a product of two binomials whenever possible.You can factor using methods like:

• Finding two numbers that multiply to \( ac \) and add to \( b \).
• Using special formulas like the difference of squares, where \( a^2 - b^2 = (a - b)(a + b) \).

In the given function \( f(x) = \frac{-4}{x^2 + 6x} \), the quadratic \( x^2 + 6x \) factors to \( x(x + 6) \). Factoring allows us to identify critical points, which are values at which the function can potentially be undefined.

Interval Notation

Interval notation is a mathematical way of writing down a range of values. It succinctly denotes where a function is defined, or represents the solution set for inequalities.In interval notation:

• Round brackets \((a, b)\) denote open intervals, where the endpoints are not included.
• Square brackets \([a, b]\) denote closed intervals, where the endpoints \( a \) and \( b \) are included.
• The union symbol \( \cup \) indicates the combination of separate intervals.

In our solution, the domain of \( f(x) = \frac{-4}{x^2 + 6x} \) is expressed as \(( -\infty, -6) \cup (-6, 0) \cup (0, \infty)\), which indicates \( x \) can be any number except \( -6 \) or \( 0 \).

Denominator Exclusion

Understanding denominator exclusion is crucial when determining the domain of a rational function. If a denominator equals zero, the function becomes undefined at that point.To find values to exclude:

• Set the denominator equal to zero and solve for \( x \).
• This solution reveals the points that must be excluded from the domain.

In the original problem \( f(x) = \frac{-4}{x^2 + 6x} \), setting \( x^2 + 6x = 0 \) and solving gives \( x = 0 \) and \( x = -6 \). Thus, the domain excludes these values, ensuring the function remains defined for all other \( x \). This process is vital for maintaining the function's integrity and determining where it can be graphically represented.