# 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 , FunctionRange may use numerical optimization to estimate the range.

# Examples

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## Basic Examples(2)

Find the range of a real function:

The range of a complex function:

## Scope(7)

Real univariate functions:

Range estimated numerically: Range over a domain restricted by conditions:

Complex univariate functions:

Real multivariate functions: Real multivariate mappings:

Range over a domain restricted by conditions:

Complex multivariate functions and mappings:

## 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 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:

All values in are attained:

Use FindInstance to show that the interval is not contained in the range of :

The value is not attained:

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 , and hence has solutions: does not belong to the range of , and hence 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 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: