analyzing relationships describe the x - values for which (a) f is increasing or decreasing, (b) ( f(x)>0 )…

analyzing relationships describe the x - values for which (a) f is increasing or decreasing, (b) ( f(x)>0 ) and (c) ( f(x)<0 ).\na. the function is increasing when ( square ) and ( square ) decreasing when ( square<x<square ).\nb. the function is greater than zero when ( square<x<square ) and ( square ).\nc. the function is less than zero when ( square ).

analyzing relationships describe the x - values for which (a) f is increasing or decreasing, (b) ( f(x)>0 ) and (c) ( f(x)<0 ).\na. the function is increasing when ( square ) and ( square ) decreasing when ( square<x<square ).\nb. the function is greater than zero when ( square<x<square ) and ( square ).\nc. the function is less than zero when ( square ).

Answer

Explanation:

Step1: Analyze increasing and decreasing intervals

A function (y = f(x)) is increasing when the slope of the tangent line is positive (the graph rises from left - to - right) and decreasing when the slope of the tangent line is negative (the graph falls from left - to - right). Looking at the graph:

  • The function (f(x)) is increasing when (x\lt0) (because as (x) moves from (-\infty) to (0), the (y) - values of the function increase) and (x > 2) (as (x) moves from (2) to (\infty), the (y) - values of the function increase).
  • The function (f(x)) is decreasing when (0\lt x\lt2) (as (x) moves from (0) to (2), the (y) - values of the function decrease).

Step2: Analyze when (f(x)>0)

The function (y = f(x)>0) when the graph of the function is above the (x) - axis. The graph of (y = f(x)) is above the (x) - axis when (- 1\lt x\lt1) (the part of the graph between (x=-1) and (x = 1) is above the (x) - axis) and (x>2) (the part of the graph for (x > 2) is above the (x) - axis).

Step3: Analyze when (f(x)<0)

The function (y = f(x)<0) when the graph of the function is below the (x) - axis. The graph of (y = f(x)) is below the (x) - axis when (x\lt - 1) (the part of the graph for (x\lt - 1)) and (1\lt x\lt2) (the part of the graph between (x = 1) and (x=2)).

Answer:

a. The function is increasing when (x\lt0) and (x > 2); decreasing when (0\lt x\lt2). b. The function is greater than zero when (-1\lt x\lt1) and (x > 2). c. The function is less than zero when (x\lt - 1) and (1\lt x\lt2).