FunctionRange
FunctionRange[f,x,y]
finds the range of the real function f of the variable x returning the result in terms of y.
FunctionRange[f,x,y,dom]
considers f to be a function with arguments and values in the domain dom.
FunctionRange[funs,xvars,yvars,dom]
finds the range of the mapping funs of the variables xvars returning the result in terms of yvars.
FunctionRange[{funs,cons},xvars,yvars,dom]
finds the range of the mapping funs with the values of xvars restricted by constraints cons.
Details and Options
- funs should be a list of functions of variables xvars.
- funs and yvars must be lists of equal lengths.
- Possible values for dom are Reals and Complexes. The default is Reals.
- If dom is Reals then all variables, parameters, constants, and function values are restricted to be real.
- cons can contain equations, inequalities, or logical combinations of these.
- The following options can be given:
-
GeneratedParameters C how to name parameters that are generated Method Automatic what method should be used WorkingPrecision Automatic precision to be used in computations - With WorkingPrecision->Automatic, FunctionRange may use numerical optimization to estimate the range.
Examples
open allclose allScope (7)
Options (2)
Method (1)
By default, the results returned by FunctionRange may not be reduced:
Use Method to specify that the result should be given in a reduced form:
WorkingPrecision (1)
By default, FunctionRange attempts to compute exact results:
With finite WorkingPrecision, slower symbolic methods are not used:
Applications (13)
Basic Applications (7)
Find the range of a real function:
All real values within the range are attained:
Find the range of a discontinuous function:
The range consists of two intervals:
Find the range of ![TemplateBox[{x}, Fibonacci]](Files/FunctionRange.en/3.png "TemplateBox[{x}, Fibonacci]"){kind=small}
over the interval 
:
Between 
and 
the plot is contained within the range:
Find the range of a complex function:
The function does not attain values 
and 
:
Compute the images of the unit disk through Möbius transformations 
and 
:
The images are a disk and a half-plane:
A function is surjective if FunctionRange gives True:
You can test surjectivity using FunctionSurjective:
A surjective function attains all values:
A function is surjective on a set of values if that set of values is contained in the function's range:
Use FindInstance to show that the interval 
is contained in the range of 
:
Confirm that 
is surjective onto 
using FunctionSurjective:
Use FindInstance to show that the interval 
is not contained in the range of 
:
Confirm that 
is not surjective onto 
using FunctionSurjective:
Solving Equations and Optimization (3)
The equation 
has solutions in the real domain of 
if and only if 
belongs to the real range of 
:
belongs to the range of ![TemplateBox[{x}, LogGamma]](Files/FunctionRange.en/26.png "TemplateBox[{x}, LogGamma]"){kind=small}
, and hence ![TemplateBox[{x}, LogGamma]=3](Files/FunctionRange.en/27.png "TemplateBox[{x}, LogGamma]=3"){kind=small}
has solutions:
does not belong to the range of ![TemplateBox[{x}, LogGamma]](Files/FunctionRange.en/29.png "TemplateBox[{x}, LogGamma]"){kind=small}
, and hence ![TemplateBox[{x}, LogGamma]=-1](Files/FunctionRange.en/30.png "TemplateBox[{x}, LogGamma]=-1"){kind=small}
has no solutions:
The equation 
has complex solutions if and only if 
belongs to the complex range of 
:
belongs to the range of 
, and hence 
has solutions:
does not belong to the range of 
, and hence 
has no solutions:
Compute the infimum and the supremum of values of a function:
You can also compute the infimum and the supremum of a function using MinValue and MaxValue:
Calculus (3)
The range of a continuous function over a connected interval must be a connected interval:
The range of a discontinuous function over a connected interval may be disconnected:
The range of a discontinuous function over a connected interval may be connected too:
If a function has a limit, that limit must belong to the closure of the function's range:
The limit may not belong to the range itself:
Estimate the value of 
the integral of ![TemplateBox[{x}, SinIntegral]](Files/FunctionRange.en/41.png "TemplateBox[{x}, SinIntegral]"){kind=small}
over the interval 
:
must be between the minimum and the maximum values in the range times the length of the interval:
Verify that the value of the integral computed using Integrate satisfies the inequalities:
is equal to the average value of the function in the interval times the length of the interval:
Properties & Relations (1)
A function is surjective if its FunctionRange is True:
Use FunctionSurjective to test whether a functions is surjective:
Possible Issues (1)
Values at isolated points at which the function is real-valued may not be included in the result:
is non-real valued for 
, except for isolated values of 
:
Real values of 
for 
may lie outside the range given by FunctionRange:
Text
Wolfram Research (2014), FunctionRange, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionRange.html.
CMS
Wolfram Language. 2014. "FunctionRange." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionRange.html.
APA
Wolfram Language. (2014). FunctionRange. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionRange.html