Problem 24 - Precalculus Essentials

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 \([-∞, 14]\) or \(x \leq 14\).

Step 1: Identify the function

The given function is \(f(x) = \sqrt{84 - 6x}\). It is important to identify that this function includes a square root.

Step 2: Constraints for square root

A square root function is only real and defined when the function inside the square root is greater than or equal to zero. I.e., \(84 - 6x \geq 0\).

Step 3: Solve the inequality

Now the equation \(84 - 6x \geq 0\) need to be solved for \(x\). This results in \(x \leq \frac{84}{6} = 14\).

Step 4: State the domain

The set of all \(x\) values that make the function real and defined is the solution to the inequality. The domain can be presented in interval notation as \([-∞, 14]\). Inequality notation is also correct: \(x \leq 14\).