Problem 33 - Intermediate Algebra

Question

Specify the domain for each of the functions. $$f(x)=\sqrt{x^{2}-3 x-18}$$

Step-by-Step Solution

Answer

The domain is \( x \in (-\infty, -3] \cup [6, \infty) \)."

Step 1: Understand the Function

The function given is a square root function, \( f(x)=\sqrt{x^2-3x-18} \). The expression inside the square root must be greater than or equal to zero for real-valued outputs in this context because the square root of a negative number is not defined in the set of real numbers.

Step 2: Set the Expression Non-negative

Set the inequality for the expression inside the square root to be greater than or equal to zero: \( x^2 - 3x - 18 \geq 0 \). We will solve this inequality next.

Step 3: Factor the Quadratic Expression

First, solve the quadratic equation \( x^2 - 3x - 18 = 0 \). Factor it as \( (x - 6)(x + 3) = 0 \). This gives us the roots \( x = 6 \) and \( x = -3 \). These roots divide the number line into intervals to check the sign of the quadratic expression.

Step 4: Test the Intervals

The roots \( x = 6 \) and \( x = -3 \) divide the number line into the intervals \(( -\infty, -3 )\), \( ( -3, 6 )\), and \(( 6, \infty )\). Test a point from each interval to determine where the inequality holds true: - Test \( x = -4 \) in \( ( -\infty, -3 ) \): \( (-4 - 6)(-4 + 3) > 0 \), true.- Test \( x = 0 \) in \( ( -3, 6 ) \): \( (-6)(3) < 0 \), false.- Test \( x = 7 \) in \( ( 6, \infty ) \): \( (7 - 6)(7 + 3) > 0 \), true.

Step 5: Include the Roots in the Domain

Since \( x = 6 \) and \( x = -3 \) make the expression equal to zero, they are included in the domain. Therefore, the inequality holds for \( x \in (-\infty, -3] \cup [6, \infty) \).

Key Concepts

Square Root Function

The square root function is a type of function that involves taking the square root of a given expression. In many mathematical contexts, the square root is defined only for non-negative numbers to ensure real-valued outputs. This is because the square root of a negative number results in an imaginary number, which is not within the typical realm of basic functions and real numbers.
When dealing with functions such as \( f(x) = \sqrt{x^2 - 3x - 18} \), it is essential to ensure the expression inside the square root is non-negative, meaning it must be greater than or equal to zero.
This consideration is crucial overall, as it determines the domain of the function—the set of all possible input values \( x \) that keep the output real and defined. The domain is specifically those values of \( x \) for which the expression under the square root is non-negative.

Quadratic Inequalities

Quadratic inequalities involve expressions of the form \( ax^2 + bx + c \geq 0 \) or \( ax^2 + bx + c \leq 0 \). Solving such inequalities is a crucial step in finding the domain of a square root function where the expression inside the square root is quadratic.
To solve the inequality \( x^2 - 3x - 18 \geq 0 \), we start by finding the roots of the corresponding quadratic equation \( x^2 - 3x - 18 = 0 \).
This is done by factoring the quadratic into \( (x - 6)(x + 3) = 0 \), which gives the roots \( x = 6 \) and \( x = -3 \). These roots provide critical points that divide the number line into sections and are key in determining where the inequality holds true, shaping the domain of the function.

Interval Testing

Interval testing, or the process of testing intervals, involves examining sections of the number line, which are divided by the roots of a quadratic equation. This step is essential in verifying where a quadratic inequality is satisfied.
For the quadratic \( x^2 - 3x - 18 \geq 0 \), the roots \( x = 6 \) and \( x = -3 \) divide the number line into intervals:

• \( (-\infty, -3) \)
• \( (-3, 6) \)
• \( (6, \infty) \)

During interval testing, select a value from each interval and substitute it into the inequality \( (x - 6)(x + 3) \). This determines whether the expression is positive or negative on each interval.
This method reveals that the inequality holds true for \( x \in (-\infty, -3] \cup [6, \infty) \), meaning these are the intervals and points where the function's expression is non-negative, thus forming the domain of the square root function.