Problem 23 - Precalculus Essentials

Question

Find the domain of each function. $$f(x)=\sqrt{24-2 x}$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\sqrt{24-2x}\) is \(x \leq 12\).

Step 1: Identify the Argument of the Square Root

In this function, the argument of the square root is \(24 - 2x\). Since the square root of a number is only defined for the number greater than or equal to zero, therefore the argument \(24 - 2x\) should also be greater than or equal to zero.

Step 2: Solve the Inequality

Solve the inequality \(24 - 2x \geq 0\) to find acceptable x values. To do this, you'll need to isolate x. When you divide or multiply by a negative number while working with inequalities, you have to flip the sign of the inequality. Here this step is as follows: 1. Add \(2x\) to both sides and subtract \(0\) from both sides: \(24 \geq 2x\)2. To isolate x, divide all terms by \(2\): \(12 \geq x\) or \(x \leq 12\)

Step 3: The Domain

The solution \(x \leq 12\) is the domain of the function. This means that the function will output a real number for any x that is less than or equal to 12.