Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 StrictlyMonotonicFunctionQ

Source Notebook

Check if a function is strictly monotonic

Contributed by: Wolfram|Alpha Math Team

ResourceFunction["StrictlyMonotonicFunctionQ"][f,x]

gives True if f is strictly monotonic with respect to x.

ResourceFunction["StrictlyMonotonicFunctionQ"][f,x,property]

restricts the test to the direction dir.

Details and Options

The direction dir can be any of Automatic, "Increasing" or "Decreasing", and defaults to Automatic. When dir is Automatic, ResourceFunction["StrictlyMonotonicFunctionQ"] returns True if f is strictly increasing or strictly decreasing.
ResourceFunction["StrictlyMonotonicFunctionQ"] expects f to be a univariate expression in terms of x, similar to what might be used in Plot.
ResourceFunction["StrictlyMonotonicFunctionQ"] returns True based on the definition of f being strictly monotonic; that is, f is strictly increasing on an interval A if for all {x,y}∈A such that x<y, it is the case that f[x]<f[y].

Examples

Basic Examples (2) 

Compute whether an expression is strictly monotonic:

In[1]:=
ResourceFunction[ "StrictlyMonotonicFunctionQ", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][x^3, x]
Out[1]=

Determine whether the expression is increasing:

In[2]:=
ResourceFunction[ "StrictlyMonotonicFunctionQ", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][x^3, x, "Increasing"]
Out[2]=

Properties and Relations (3) 

Note that a function with regions where it is constant is not strictly increasing:

In[3]:=
f = Piecewise[{{x, x <= 0}, {0, 0 < x < 1}, {x - 1, x > 1}}];
In[4]:=
ResourceFunction[ "StrictlyMonotonicFunctionQ", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][f, x]
Out[4]=

For more information on the monotonicity of the input expression, use the resource function FunctionMonotonicity:

In[5]:=
ResourceFunction["FunctionMonotonicity"][f, x]
Out[5]=

Such a curve is only monotonic, not strictly monotonic:

In[6]:=
ResourceFunction["MonotonicFunctionQ"][f, x]
Out[6]=

Publisher

Wolfram|Alpha Math Team

Version History

  • 2.0.0 – 23 March 2023
  • 1.0.0 – 29 September 2020

Related Resources

Author Notes

To view the full source code for StrictlyMonotonicFunctionQ, run the following code:

In[1]:=
FileNameJoin[ ReplacePart[ FileNameSplit[FindFile["ResourceFunctionHelpers`"]], -1 -> "FunctionMonotonicityConcavity.wl"]] // SystemOpen

License Information