Question

select the correct answer. consider the piecewise function shown on the graph, which is composed of polynomial, constant, and exponential pieces. over which interval of the domain is the function increasing? a. (-∞, -4) b. (1, 4) c. (-∞, 1) d. (-∞, 4)

Explanation:

Step1: Recall the definition of an increasing function

A function \(y = f(x)\) is increasing on an interval if, for any two points \(x_1\) and \(x_2\) in the interval with \(x_1

Step2: Analyze the given graph

• For the interval \((-\infty,-4)\):

As we move from left - to - right (i.e., as \(x\) increases) on the interval \((-\infty,-4)\), the \(y\) - values of the function are increasing.

• For the interval \((1,4)\):

As we move from left - to - right on the interval \((1,4)\), the \(y\) - values of the function are decreasing (since it is an exponential - like decay part of the piece - wise function).

• For the interval \((-\infty,1)\):

The function has a non - increasing part (the constant part for \(x\in[-4,0]\)). When \(x\in[-4,0]\), \(y\) is constant (\(y = 4\)), so the function is not increasing for all \(x\in(-\infty,1)\).

• For the interval \((-\infty,4)\):

The function has non - increasing parts (the constant part for \(x\in[-4,0]\) and the decreasing part for \(x\in(0,4)\)).

Answer:

A. \((-\infty,-4)\)