Question

find the vertex of the given function. f(x) = |x - 5| + 10 the vertex is at (5, 10). complete what are the domain and range of the function? ① domain: (-∞, ∞); range: f(x) ≥ -5 ② domain: (-∞, ∞); range: f(x) ≥ 10 ③ domain: x ≥ 5; range: f(x) ≥ 10 ④ domain: f(x) ≥ 5; range: (-∞, ∞) done

Explanation:

Step1: Analyze the domain of the absolute value function

The function is \( f(x) = |x - 5| + 10 \). The absolute value function \( |x - a| \) is defined for all real numbers \( x \), so there are no restrictions on \( x \). Thus, the domain is all real numbers, which is \( (-\infty, \infty) \).

Step2: Analyze the range of the absolute value function

The absolute value \( |x - 5| \) is always non - negative, i.e., \( |x - 5| \geq 0 \) for all real numbers \( x \). When we add 10 to both sides of the inequality, we get \( |x - 5|+ 10\geq0 + 10 \), so \( f(x)=|x - 5| + 10\geq10 \). So the range is \( f(x)\geq10 \).

Answer:

B. domain: \( (-\infty, \infty) \); range: \( f(x)\geq10 \)