Increasing And Decreasing Functions

Even And Odd FunctionsComposition Of Functions

Monotonicity describes the trend of a function as the input moves from left to right. Determining where functions increase or decrease is one of the foundational applications of derivatives.

Quick Reference

Function Behavior Condition for x 1 < x 2 Graphical Trend
Increasing f ( x 1 ) < f ( x 2 ) Rises from left to right
Decreasing f ( x 1 ) > f ( x 2 ) Falls from left to right
Constant f ( x 1 ) = f ( x 2 ) Horizontal straight line

Definitions of Monotonicity

Let f be a function defined on an interval I.

  1. Increasing Function: A function f is increasing on an interval I if: f ( x 1 ) < f ( x 2 ) whenever x 1 < x 2  in  I
    Graph of an increasing function rising from left to right
    An increasing curve rises as x increases.
  2. Decreasing Function: A function f is decreasing on an interval I if: f ( x 1 ) > f ( x 2 ) whenever x 1 < x 2  in  I
    Graph of a decreasing function falling from left to right
    A decreasing curve falls as x increases.
  3. Constant Function: A function f is constant on an interval I if: f ( x 1 ) = f ( x 2 ) for all  x 1 , x 2  in  I
    Graph of a horizontal constant function
    A constant function has a horizontal line graph.
Even And Odd FunctionsComposition Of Functions
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