`<piecewiseFunction>`

A function defined piecewise from sub-functions on intervals

<piecewiseFunction> is a Graphical component that renders a piecewise defined function that can be evaluated and graphed in the same manner as a <function> component.

In order to evaluate a piecewiseFunction, the function attribute symbolic must be set to false, to force numerical evaluation.

Disclaimer: currently, piecewise functions cannot be used with the <derivative> component. Individual <function> children need to be differentiated independently. Open and closed endpoints must also be graphed as independent components (see <endpoint>.)

Attributes and Properties

Attributes for `<piecewiseFunction>`

Number display (5)

avoidScientificNotation

boolean. Default value: false. Whether to render numbers in full decimal form rather than scientific notation.

displayDecimals

integer. Default value: 2. Number of decimal places to display when rendering this number.

displayDigits

integer. Default value: 3. Number of significant digits to display when rendering this number.

displaySmallAsZero

number. Default value: 1e-14. Threshold below which numbers are displayed as zero.

padZeros

boolean. Default value: false. Whether to pad displayed numbers with trailing zeros to fill the requested digits/decimals.

Labels (1)

labelIsName

boolean. Default value: false. Whether to use this component's name as its rendered label.

Other (13)

applyStyleToLabel

boolean. Default value: false. Whether to apply the function's style to its label.

domain

intervalList. Restriction of the function's domain to a list of intervals.

expand

boolean. Default value: false. Whether to expand the function's formula.

labelPosition

keyword. Position of the function's label.

Value	Description
`upperRight` (default)	Place the label above and to the right of the function curve.
`upperLeft`	Place the label above and to the left of the function curve.
`lowerRight`	Place the label below and to the right of the function curve.
`lowerLeft`	Place the label below and to the left of the function curve.
`top`	Place the label directly above the function curve.
`bottom`	Place the label directly below the function curve.
`left`	Place the label directly to the left of the function curve.
`right`	Place the label directly to the right of the function curve.

layer

number. Default value: 0. Z-order layer index for stacking the function on a graph.

nearestPointAsCurve

boolean. Default value: false. Whether nearest-point queries should treat the function as a curve in the plane rather than a graph y = f(x).

simplify

keyword. Level of simplification applied to the function's formula.

Value	Description
`none`	No simplification is applied.
`full` (default)	Fully simplify the function's formula.
`numbers`	Simplify numeric subexpressions only, leaving symbolic structure intact.
`numbersPreserveOrder`	Like `numbers`, but does not reorder commutative operands.
`normalizeOrder`	Reorder commutative operands into a canonical form without simplifying values.

symbolic

boolean. Default value: true. Whether the function should be evaluated symbolically rather than numerically.

throughSlopes

mathList. Slopes the interpolated function should have at each through-point.

variable

_variableName. Name of the function's single input variable.

variables

[ _variableNameList ]. Names of the function's input variables.

xscale

number. Default value: 1. Scale factor used along the x-axis.

yscale

number. Default value: 1. Scale factor used along the y-axis.

Common to all components (9)

copy

reference. Create an independent copy of another component by reference. Enter a references a $name.

disabled

boolean. Default value: false. Whether this component is disabled and cannot be interacted with.

extend

reference. Extend another component by reference, inheriting its children and attributes. Enter a reference as $name.

fixed

boolean. Default value: false. Whether this component's value is fixed and cannot be modified.

fixLocation

boolean. Default value: false. Whether this component's location is fixed (preventing it from being moved while still allowing other modifications).

hide

boolean. Default value: false. Whether to hide this component from the rendered output.

isResponse

boolean. Default value: false. Whether this component is treated as a response for the purposes of assessment.

name

text. The name used to reference this component from elsewhere in the document.

styleNumber

integer. Default value: 1. The style number used to select this component's visual styling from the available style definitions.

Properties for `<piecewiseFunction name="p">`

Number display (5)

$p.avoidScientificNotation

boolean. Whether to render numbers in full decimal form rather than scientific notation.

$p.displayDecimals

integer. Number of decimal places to display when rendering this number.

$p.displayDigits

integer. Number of significant digits to display when rendering this number.

$p.displaySmallAsZero

number. Threshold below which numbers are displayed as zero.

$p.padZeros

boolean. Whether to pad displayed numbers with trailing zeros to fill the requested digits/decimals.

Labels (2)

$p.label

label. The label rendered with this component.

$p.labelIsName

boolean. Whether to use this component's name as its rendered label.

Other (41)

$p.applyStyleToLabel

boolean. Whether to apply the function's style to its label.

$p.backgroundColor

text. Human-readable name for this component's background color, derived from the active style and theme.

$p.disabled

boolean. Whether this component is disabled and cannot be interacted with.

$p.expand

text. Whether to expand the formula.

$p.expandSpecified

boolean. Whether to expand the function's formula.

$p.extrema

[ number ]. Local extrema (minima + maxima) of the function.

$p.extremumLocations

[ number ]. The x-coordinates of the function's local extrema.

$p.extremumValues

[ number ]. The values of the function's local extrema.

$p.fixed

boolean. Whether this component's value is fixed and cannot be modified.

$p.fixLocation

boolean. Whether this component's location is fixed (preventing it from being moved while still allowing other modifications).

$p.formula

math. The piecewise formula of the function.

$p.globalInfimum

[ number ]. The global infimum of the function.

$p.globalMaximum

[ number ]. The global maximum of the function (if attained).

$p.globalMinimum

[ number ]. The global minimum of the function (if attained).

$p.globalSupremum

[ number ]. The global supremum of the function.

$p.hidden

boolean. Whether this component is hidden from the rendered output.

$p.labelPosition

text. Position of the function's label.

$p.latex

latex. The piecewise function rendered as a LaTeX string.

$p.layer

number. Z-order layer index for stacking the function on a graph.

$p.maxima

[ number ]. Local maxima of the function.

$p.maximumLocations

[ number ]. The x-coordinates of the function's local maxima.

$p.maximumValues

[ number ]. The values of the function's local maxima.

$p.minima

[ number ]. Local minima of the function.

$p.minimumLocations

[ number ]. The x-coordinates of the function's local minima.

$p.minimumValues

[ number ]. The values of the function's local minima.

$p.numExtrema

number. Number of local extrema (minima + maxima) of the function.

$p.numInputs

integer. Number of input variables.

$p.numMaxima

number. Number of local maxima of the function.

$p.numMinima

number. Number of local minima of the function.

$p.numOutputs

integer. Number of output values.

$p.simplify

text. Level of simplification applied when displaying the formula.

$p.styleDescription

text. A textual description of the function's style.

$p.styleDescriptionWithNoun

text. Style description including the word "function".

$p.symbolic

boolean. Whether the function is treated symbolically.

$p.text

text. The function rendered as a plain text string.

$p.textColor

text. Human-readable name for this component's text color, derived from the active style and theme.

$p.textStyleDescription

text. Human-readable description of this component's text styling (color and any background color).

$p.variable

_variableName. The function's first input variable name.

$p.variables

[ _variableName ]. The names of the function's input variables.

$p.xscale

number. Scale factor used along the x-axis.

$p.yscale

number. Scale factor used along the y-axis.

Common to all components (4)

$p.doenetML

text. The DoenetML source code that produced this component.

$p.hide

boolean. Whether to hide this component from the rendered output.

$p.isResponse

boolean. Whether this component is treated as a response for the purposes of assessment.

$p.styleNumber

integer. The style number used to select this component's visual styling from the available style definitions.

Examples

Example: A simple three-part `<piecewiseFunction>`

The <piecewiseFunction> component is defined by assigning <function> children for each distinct region of the function’s domain. The named <piecewiseFunction> can then be evaluated in the same manner as the <function> component using the $$functionName(input) evaluation reference.

Note: When using a <piecewiseFunction> for evaluation, it is necessary to include the function attribute symbolic="false" to force numerical evaluation.

Example: Adding endpoints graphically to a `<piecewiseFunction>`

Doenet only evaluates functions at points strictly in their specified domains. Therefore, in order to locate endpoints of open intervals for graphing, it may be necessary to define a set of base functions with unrestricted domains. In the example shown, the child functions themselves (and therefore the endpoint evaluations) are trivial, but the same method will work for more complex functions as well.

These predefined functions can then be copied into the definition of the <piecewiseFunction> using the extend attribute, and the precise domains can be specified for each. This allows for the single <piecewiseFunction> to then be used for

graphing the linear component of the function,
for correctly rendering the equation of the piecewise function outside the graph, and
for correct mathematical evaluation at all points in the domain.

Note: When using a <piecewiseFunction> for evaluation, it is necessary to include the function attribute symbolic="false" to force numerical evaluation.

Example: How domains are interpreted

Domains may be specified for the parent <piecewiseFunction>, and also for all individual child functions. When two domains overlap, the previously defined <function> governs. Finally, when no domain is specified for a given function, that function assumes the domain not occupied by all of the other defined functions.

Note: When using a <piecewiseFunction> for evaluation, it is necessary to include the function attribute symbolic="false" to force numerical evaluation.

Attribute Examples

Attribute Example: domain

The domain attribute can be used to specify an overarching domain for the <piecewiseFunction>.

Attribute Example: variable/ variables

The variables attribute changes the overarching variable used for the <piecewiseFunction>, but individual <function> variables may differ. At this time (update forthcoming), the child functions will still look for a default independent variable of $x$ .

Property Examples

Property Example: minima/ maxima

Local maxima and minima of the <piecewiseFunction> are rendered with the maxima/minima properties.

periodicSet pluralize

<piecewiseFunction>