Question

this graph shows a function. for which intervals is the function decreasing? select all that apply. 5<x<8 4<x<9 2<x<6 0<x<3 1<x<4 3<x<5

Explanation:

Step1: Recall the definition of a decreasing function

A function \(y = f(x)\) is decreasing on an interval \((a,b)\) if for any \(x_1,x_2\in(a,b)\) with \(x_1

Step2: Analyze each interval

• For the interval \(0 < x<3\):

As \(x\) increases from \(0\) to \(3\), the \(y\) - values of the function decrease. For example, when \(x = 0,y\approx7\) and when \(x = 3,y = 2\).

• For the interval \(1 < x<4\):

Since \(1

• For the interval \(2 < x<6\):

From \(x = 2\) to \(x = 3\) the function is decreasing, from \(x = 3\) to \(x = 5\) the function is increasing, and from \(x = 5\) to \(x = 6\) the function is decreasing. So, \(2 < x<6\) is not a single - valued decreasing interval.

• For the interval \(3 < x<5\):

As \(x\) increases from \(3\) to \(5\), the \(y\) - values of the function increase. So, it is an increasing interval.

• For the interval \(4 < x<9\):

As \(x\) increases from \(4\) to \(9\), the \(y\) - values of the function first decrease (from \(x = 4\) to \(x = 8\)) and then remain constant (from \(x = 8\) to \(x = 9\)). So, \(4 < x<9\) is not a decreasing interval.

• For the interval \(5 < x<8\):

As \(x\) increases from \(5\) to \(8\), the \(y\) - values of the function decrease.

Answer:

\(0 < x<3\) and \(5 < x<8\)