Question

given the function ( h(x) ), over which of the following intervals is ( h ) only increasing? ( -3 lt x lt 3 ) ( 1 lt x lt 4 ) ( -1 lt x lt 4 ) ( -3 lt x lt 1 )

Explanation:

Step1: Understand the concept of increasing function

A function \(y = h(x)\) is increasing if as \(x\) increases, \(y\) also increases. On a graph, this means the function has a positive - slope.

Step2: Analyze each interval

• For the interval \(-3 < x < 3\):

The function first decreases (negative slope) and then increases (positive slope) and then decreases (negative slope) again. So, it is not only increasing in this interval.

• For the interval \(1 < x < 4\):

The function is decreasing (negative slope) in this interval.

• For the interval \(-1 < x < 4\):

The function first increases (from \(x=-1\) to \(x = 1\)) and then decreases (from \(x = 1\) to \(x=4\)). So, it is not only increasing in this interval.

• For the interval \(-3 < x < 1\):

Looking at the graph, as \(x\) moves from \(-3\) to \(1\), the \(y\) - values of the function \(h(x)\) are continuously increasing. The slope of the function in this interval is positive.

Answer:

$-3 < x < 1$