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Increasing and Decreasing Functions

x value intervals where the y values are moving up or down.

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Increasing and Decreasing

Functions are increasing in a given interval (a,d) when any two numbers (called b and c) in the interval where b<c has the property f(b)≤f(c).  A review of interval notation can be found here.

Which of the basic functions are increasing over their domain?

Decreasing functions logically have the opposite definition.  If in a given interval (a,d) any two numbers b and c have the property f(b)≥f(c) where b<c, then the function is decreasing over (a,d).

Which of the basic functions are decreasing over their domain?

Tangent lines are lines that touch the curve of a graph at only one place.  An alternative definition of an increasing interval of a function is an interval where slope of the tangent line is positive.  By this logic, a decreasing interval can also be defined as an interval where the slope of the tangent line is negative.  Learn more about tangent lines here.

State the increasing and decreasing intervals of the following function:

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 Answers below

1. The identity function, the cubic function, the square root function, the exponential function, the logarithmic function, the sin and cos functions, the logistic function

2. The reciprocal function

3. Inc: (- ∞,-3) U (3, ∞)  Dec: (-3,-2) U (-2,2) U (2,3)

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