Question

what are the domain and range of f(x) = 2|x - 4|?
option 1: domain: x ≤ 2; range: (-∞,∞)
option 2: domain: (-∞,∞); range: f(x) ≥ 0
option 3: domain: (-∞,∞); range: f(x) ≤ 0
option 4: domain: x ≥ 2; range: (-∞,∞)

Explanation:

Step1: Analyze Domain of Absolute Value Function

The function is \( f(x) = 2|x - 4| \). Absolute value functions are defined for all real numbers \( x \), so the domain is \( (-\infty, \infty) \).

Step2: Analyze Range of Absolute Value Function

The absolute value \( |x - 4| \) is always non - negative, i.e., \( |x - 4|\geq0 \). When we multiply this by 2 (a positive number), the inequality direction remains the same. So \( 2|x - 4|\geq0 \), which means \( f(x)\geq0 \). So the range is \( f(x)\geq0 \).

Answer:

domain: \((-\infty,\infty)\); range: \(f(x)\geq0\) (the second option)