Questions: Identify the domain of the following function: f(x) = sqrt((x^2 - 8x)/4) (-8,0) (-∞,-8) ∪ (0, ∞) (0,8) (-∞, 0) ∪ (8, ∞)

Solution Steps

The function is \( f(x) = \sqrt{\frac{x^{2} - 8x}{4}} \). For the square root to be defined, the expression inside the square root must be non-negative: \[ \frac{x^{2} - 8x}{4} \geq 0. \]

Multiply both sides of the inequality by 4 to eliminate the denominator: \[ x^{2} - 8x \geq 0. \]

Factor the quadratic expression: \[ x(x - 8) \geq 0. \]

The critical points are the values of \( x \) that make the expression equal to zero: \[ x = 0 \quad \text{and} \quad x = 8. \]

The critical points divide the number line into three intervals:

1. \( x < 0 \)
2. \( 0 < x < 8 \)
3. \( x > 8 \)

Test a value from each interval in the inequality \( x(x - 8) \geq 0 \):

• For \( x < 0 \), choose \( x = -1 \): \[ (-1)(-1 - 8) = (-1)(-9) = 9 \geq 0. \]
• For \( 0 < x < 8 \), choose \( x = 4 \): \[ (4)(4 - 8) = (4)(-4) = -16 < 0. \]
• For \( x > 8 \), choose \( x = 9 \): \[ (9)(9 - 8) = (9)(1) = 9 \geq 0. \]

The inequality \( x(x - 8) \geq 0 \) holds true for: \[ x \leq 0 \quad \text{or} \quad x \geq 8. \]

The domain of the function is: \[ (-\infty, 0] \cup [8, \infty). \]

Final Answer