91Ó°ÊÓ

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 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.

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.