identify the interval(s) on which the function is constant.\na. (3,∞)\nb. (-∞,0)\nc. (-∞,-1) and (3,∞)\nd…

Explanation:

Step1: Recall the definition of a constant function

A function (y = f(x)) is constant on an interval if for any (x_1,x_2) in that interval, (f(x_1)=f(x_2)). Graphically, a constant - function has a horizontal line segment.

Step2: Analyze each option

• Option A: For the interval ((3,\infty)), as (x) increases in ((3,\infty)), the (y) - value remains (y = 4).
• Option B: For the interval ((-\infty,0)), when (x\in(-2,0)), the function is not constant (it has a non - zero slope in ((- 1,0))).
• Option C: For the interval ((-\infty,-1)), the function is constant ((y = 1)), but for (x\in(3,\infty)) it is also constant ((y = 4)), but the question asks for the interval(s) on which the function is constant. The function is not constant over the union ((-\infty,-1)\cup(3,\infty)) as a whole (since the constant values (y = 1) and (y = 4) are different).
• Option D: For the interval ((-1,0)), the function has a positive slope (it is increasing), so it is not constant.

Answer:

A. ((3,\infty))