Problem 70 - College Algebra

Question

Find the domain of each function. $$ f(x)=\sqrt{84-6 x} $$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\sqrt{84-6x}\) is \(x \leq 14\).

Step 1: Setting the Inequality

Set the expression inside the square root to be greater than or equal to 0: \(84 - 6x \geq 0\)

Step 2: Isolate Variable x

In order to isolate x, we'll first subtract 84 from both sides of the inequality: \(-6x \geq -84\). Then, in the next step, take care to remember that when you divide or multiply both sides of an inequality by a negative number, the inequality reverses direction. Dividing both sides by -6 giving us \(x \leq 14\).

Step 3: Write the Solution

Having isolated x, the solution states that the domain of the function \(f(x)=\sqrt{84-6x}\) is \(x \leq 14\). This means that the function is defined for all x-values that are less than or equal to 14.

Key Concepts

Understanding the Square Root Function

At the core of the original exercise is the square root function. This function involves finding the square root of a given expression, which can be a number, a variable, or a more complex expression, like we have here with \( f(x) = \sqrt{84-6x} \).

For any square root function, remember that the square root of a negative number is not a real number. That's why when dealing with square roots, we need to make sure that the expression inside the square root (called the radicand) is non-negative. This characteristic is fundamental in determining the domain of square root functions.

• A square root function is only defined when the radicand is \( \geq 0 \).
• The radicand must not be negative to ensure the output of the square root function is a real number.

Getting comfortable with these aspects of square root functions will assist in tackling any problems involving them more effectively.

Solving Inequalities

Solving inequalities is a crucial skill when working with square root functions and determining their domains. In our exercise, the inequality \(84 - 6x \geq 0\) plays a key role. To solve this, it's essential to remember a few principles about inequalities.

• When you solve inequalities, treat them similar to equations, with an important exception: when you multiply or divide by a negative number, the direction of the inequality sign reverses.
• In our context, dividing both sides of \(-6x \geq -84\) by \(-6\) reversed the inequality from \( \geq \) to \( \leq \).

Solving inequalities correctly allows us to pinpoint the range of x-values for which the square root function is defined, thereby defining the function's domain.

Exploring Domain Restrictions

Understanding the domain of a function is about identifying all the possible input values (x-values) that will produce real output values. In the function \( f(x) = \sqrt{84 - 6x} \), we needed to find which x-values keep the radicand non-negative.

• The result from solving the inequality \( x \leq 14 \) defines our domain.
• This means any x-value less than or equal to 14 will keep \( 84 - 6x \geq 0 \), ensuring the square root presents a real number.

Domains may be restricted due to various reasons like division by zero or square root of negative numbers, making it key to always check the limitations on x-values. Grasping domain restrictions allows us to understand where and how functions behave or do not operate.