Problem 24 - College Algebra 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 \(x \leq 14\). This means that the function is defined for all \(x\) less than or equal to 14.

Step 1: Identify the Inner Function

Look at the inner function, which is \(84-6x\). This is inside the square root.

Step 2: Set the Inner Function to Greater Than or Equal to Zero

For the value inside the square root to be real, it must be greater than or equal to 0, which gives the inequality \(84-6x \geq 0\)

Step 3: Solve the Inequality for x

To find the values of \(x\) that satisfy the inequality, isolate \(x\) on one side of the equation. Start by subtracting 84 from both sides, which gives \(-6x \geq -84\). Then divide both sides by -6. Remember that dividing or multiplying an inequality by a negative number reverses the inequality sign, resulting in \(x \leq 14\)