Problem 74 - A Graphical Approach to College Algebra

Question

Determine the domain of each function. Do not use a calculator. $$f(x)=\sqrt{9 x+18}$$

Step-by-Step Solution

Answer

Domain: \([-2, \infty)\)

Step 1: Identify the Expression Inside the Square Root

In the function \( f(x) = \sqrt{9x + 18} \), the expression inside the square root is \( 9x + 18 \). Square root functions require the expression inside to be non-negative.

Step 2: Set Up Inequality for Non-Negative Expression

To ensure the value inside the square root is non-negative, set up the inequality: \( 9x + 18 \geq 0 \). This inequality will determine the values of \( x \) that keep the expression non-negative.

Step 3: Solve the Inequality

Start solving the inequality \( 9x + 18 \geq 0 \) by subtracting 18 from both sides, giving \( 9x \geq -18 \). Next, divide both sides by 9 to isolate \( x \), resulting in \( x \geq -2 \).

Step 4: Write the Domain in Interval Notation

The solution \( x \geq -2 \) indicates that the domain includes all \( x \) values greater than or equal to -2. In interval notation, the domain is \([ -2, \infty )\).

Key Concepts

Inequalities

When dealing with inequalities, it's essential to understand what they represent. Inequalities indicate the relationship between two expressions that are not necessarily equal. In the context of finding the domain of a function, especially with radical expressions, the inequality helps us determine the set of possible values for the variable.

• Understanding Inequalitites: An inequality uses symbols like \(<\), \(>\), \(\leq\), and \(\geq\) to show the relationship between expressions.
• Non-negative Requirement: For functions involving square roots, the expression inside the square root must be greater than or equal to zero. This prevents us from dealing with imaginary numbers, which aren't considered in real number functions.

To solve an inequality, follow the normal algebraic steps to isolate the variable. For example, in the inequality \(9x + 18 \geq 0\), subtract 18 from both sides and divide by 9 to solve for \(x\). This gives \(x \geq -2\). This solution means any value of \(x\) equal to or greater than \(-2\) satisfies the inequality.

Radical Expressions

A radical expression involves roots, such as square roots, cube roots, etc. When you see a square root, you are dealing with a radical expression with an index of two.

• Key Properties: Radical expressions often require the expression inside to meet specific conditions, like being non-negative in square roots, to ensure real number results.
• Simplifying: Simplifying radical expressions, when possible, makes them easier to work with. However, understanding the conditions for the expression is crucial when determining domains.

In our function, \(f(x) = \sqrt{9x + 18}\), it is necessary for \(9x + 18\) to be non-negative, as we can't take the square root of a negative number in the set of real numbers. Solving \(9x + 18 \geq 0\) ensures that we're working within real numbers.

Interval Notation

Interval notation is a shorthand method to describe a set of numbers. It is highly useful in expressing the domain of a function, as it quickly conveys the range of possible values.

• Components: An interval is written with brackets and parentheses. A square bracket \([\) or \(]\) represents that the endpoint is included in the interval. A parenthesis \((\) or \()\) signifies the endpoint is not included.
• Example: For \(x \geq -2\), the interval notation is \([-2, \infty)\), indicating that \(x\) starts at \(-2\) and goes to infinity, including \(-2\) but not a numeric value for infinity.

Interval notation is a compact and clear way to show the domain of a function. It's especially convenient in written communication, avoiding lengthy descriptions.