select the correct answer. on what interval is the function h(x) = |x - 5| + 2 increasing? a. (2, ∞) b. (5…

select the correct answer. on what interval is the function h(x) = |x - 5| + 2 increasing? a. (2, ∞) b. (5, ∞) c. (-∞, 2) d. (-∞, 5)

select the correct answer. on what interval is the function h(x) = |x - 5| + 2 increasing? a. (2, ∞) b. (5, ∞) c. (-∞, 2) d. (-∞, 5)

Answer

Explanation:

Step1: Rewrite absolute - value function

The absolute - value function (y = |x - 5|) can be rewritten as a piece - wise function: (y=\begin{cases}x - 5, & x\geq5\-(x - 5), & x<5\end{cases}). So, (h(x)=\begin{cases}x - 5+2=x - 3, & x\geq5\-(x - 5)+2=-x + 7, & x<5\end{cases}).

Step2: Analyze the slope of each part

For (h(x)=x - 3) when (x\geq5), the slope (m = 1>0), which means the function is increasing on the interval ([5,\infty)). For (h(x)=-x + 7) when (x<5), the slope (m=-1<0), which means the function is decreasing on the interval ((-\infty,5)).

Answer:

B. ((5,\infty))