Question

for which interval(s) is the function increasing? (2,10) (-2,6) (-5,0) (0,0) (5,0) (-8,-4) (2,5) (-∞,-8) (-8,-2) (-2,0) (5,∞) (0,2)

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 each interval

• Interval \((2,5)\):

The function is decreasing on \((2,5)\) because as \(x\) increases from \(2\) to \(5\), \(y\) decreases.

• Interval \((-\infty,-8)\):

The function is decreasing on \((-\infty,-8)\) because as \(x\) increases (moving from left - to - right in the interval \((-\infty,-8)\)), \(y\) decreases.

• Interval \((-8,-2)\):

As \(x\) increases from \(-8\) to \(-2\), \(y\) increases. For example, when \(x=-8,y = - 4\) and when \(x=-2,y = 6\). So the function is increasing on \((-8,-2)\).

• Interval \((-2,0)\):

The function is decreasing on \((-2,0)\) because as \(x\) increases from \(-2\) to \(0\), \(y\) decreases (from \(y = 6\) to \(y = 0\)).

• Interval \((5,\infty)\):

As \(x\) increases from \(5\) to \(\infty\), \(y\) increases. So the function is increasing on \((5,\infty)\).

• Interval \((0,2)\):

As \(x\) increases from \(0\) to \(2\), \(y\) increases (from \(y = 0\) to \(y = 10\)). So the function is increasing on \((0,2)\).

Answer:

\((-8,-2)\), \((0,2)\), \((5,\infty)\)