contextPiecewiseFunction.pl - Allow usage of piecewise functions.

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NAME

contextPiecewiseFunction.pl - Allow usage of piecewise functions.

DESCRIPTION

This file implements a context in which piecewise-defined functions can be specified by students and problem authors. To use it, add

        loadMacros("contextPiecewiseFunction.pl");

and then use

        Context("PiecewiseFunction");

to select the context for piecewise functions. There are several ways to produce a piecewise function. For example:

        $f = Compute("x if x >= 0 else -x");
        $f = Compute("x if x >= 0 else -x if x < 0");
        $f = Formula("x+1 if x > 2 else 4 if x = 2 else 1-x");
        $f = PiecewiseFunction("x^2 if 1 < x <= 2 else 2x+1");
        $f = PiecewiseFunction("1 < x <= 2" => "x^2", "2x+1");
        $f = PiecewiseFunction("(1,2]" => "x^2", "2x+1");
        $f = PiecewiseFunction(Interval("(1,2]") => "x^2", "2x+1");

You can use either Compute() or Formula() interchangeably to convert a string containing "if" and "else" to the corresponding PiecewiseFunction. The PiecewiseFunction() constructor can also do this, or you can pass it a list of interval=>formula pairs that specify the various branches. If there is an unpaired final formula, it represents the "otherwise" portion of the function (the formula to use of the input is not in any of the given intervals).

Note that you can use Inveral, Set, or Union objects in place of the intervals in the specification of a piecewise function.

The PiecewiseFunction object TeXifies using a LaTeX "cases" environment, so you can use these objects to produce nice output even if you are not asking a student to enter one. For example:

        Context("PiecewiseFunction");
        
        $f = Formula("1-x if x > 0 else 4 if x = 0 else 1+x if x < 0");
        $a = random(-2,2,.1);
        
        Context()->texStrings;
        BEGIN_TEXT
        If \[f(x)=$f\] then \(f($a)\) = \{ans_rule(20)\}.
        END_TEXT
        Context()->normalStrings;
        
        ANS($f->eval(x=>$a)->cmp);

Normally when you use a piecewise function at the end of a sentence, the period is placed at the end of the last case. Since

        \[ f(x) = $f \].

would put the period centered at the right-hand side of the function, this is not what is desired. To get a period at the end of the last case, use

        \[ f(x) = \{$f->with(final_period=>1)\} \]

instead.