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

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)

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)

Answer

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<x_2), we have ( f(x_1)<f(x_2)). Graphically, this means the function rises as we move from left - to - right.

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))