Problem 23 - College Algebra 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: Define the Condition

Any real number under a square root must be greater than or equal to zero. Hence, for the function \(f(x) = \sqrt{24 - 2x}\), the expression \( 24 - 2x \) must be greater than or equal to zero.

Step 2: Solve the Equation

Solve the inequality \(24 - 2x \geq 0\).\nSubtract 24 from both sides to get \(-2x \geq -24\).\nThen, divide by -2 on both sides. However, remember that when you multiply or divide an inequality by a negative number, the direction of the inequality changes, giving you \(x \leq 12 \). So any 'x' less than or equal to 12 falls within the domain.

Step 3: State the Final Domain

The domain of the function \(f(x) = \sqrt{24 - 2x}\) is \(x \leq 12\). This means that 'x' can be any real number less than or equal to 12.