Question

What is the meaning of monotonically increasing function?

Answer 1

Answer:

Introduction

The groups of monotonically increasing and monotonically decreasing functions have some special properties. A monotonically increasing function is one that increases as x

does for all real x. A monotonically decreasing function, on the other hand, is one that decreases as x increases for all real x

. In particular, these concepts are helpful when studying exponential and logarithmic functions.

Monotonically Increasing Functions

The graphs of exponential and logarithmic functions will be crucial here. From them we can see a general rule:

When a Function is not Monotonically Increasing or Decreasing

There are some functions that are not monotonically increasing nor monotonically decreasing. There are an infinite number of these functions, and they belong to many different groups.

Main Group 1: Constant Functions

These are straight lines, so they are not decreasing or decreasing.

Main Group 2: Absolute Value Functions

Functions surrounded by an absolute value sign are always nonnegative, but then all non-constant functions of this type will have a minimum. Therefore the function will alternate between increasing and decreasing as x

increases.

Main Group 3: Trigonometric Functions

Consider basic trigonometric functions such as sin(x)

, which move up and down, and thus do not exclusively increase or decrease.

Main Group 4: Functions with Discontinuities

A function cannot increase or decrease over any type of discontinuity, especially when the discontinuity is caused by an undefined value (i.e. x=0

in f(x)=1x

).

Monotonically increasing and decreasing graphs can be identified by graphs, but this is not a very rigorous method. Still, it is good for students who do not have any calculus background. There are methods containing much more detail and rigor that involve calculus, related to the rate of change of the function as x

changes.