Piecewise-Defined Functions

You can define functions of one variable that are described by different expressions on different parts of their domain. These functions are referred to as piecewise-defined functions, case functions or multicase functions. Most of the operations introduced in calculus are now supported for piecewise-defined functions. You can evaluate, plot, differentiate, and integrate piecewise-defined functions in Scientific Notebook.

$\blacktriangleright$ To define a piecewise-defined function

1.
Type an expression of the form f (x) =.

2.
From the brackets list itbpF0.2992in0.2992in0.0692infences.wmf, choose a curly brace itbpF0.1989in0.2992in0.0692inFigure for the left bracket and the dashed vertical line (or null bracket) itbpF0.1989in0.2992in0.0692inFigure for the right bracket.

3.
Click the Matrix button on the Math toolbar, or from the Insert menu, choose Matrix.

4.
Set the number of rows equal to the number of ``pieces.''

5.
Set the number of columns to 3.

6.
Choose OK.

7.
Type function values in the first column.

8.
Type ``if'' in the second column (in text or mathematics mode).

9.
Type the range condition in the third column, beginning with smallest values of the variable.

Note the strict conditions concerning the definition of piecewise-defined functions. Such functions must be specified by a three-column matrix with at least two rows. The function values must be in the first column, and the range conditions must be in the third column. Also, the matrix must be enclosed by a left brace and a ``null'' right delimiter, as in the following examples f and g. (The null right delimiter appears as a red dashed line on the screen when Matrix Lines is checked on the View menu, and does not appear when the document is printed or when Matrix Lines is not checked.)


\begin{example} The function $f$\ has three \lq\lq cases'' or range conditions on th... ... 6-t & \text{if} & 3\leq t \end{array} \right. $ \end{itemize} \end{example}

With the insertion point in expressions such as the preceding ones (or with the entire equation selected with the mouse), click the New Definition button on the Compute toolbar, or go to the Define submenu and choose New Definition. Choose Evaluate to get results such as

f (- 1) =  1    f ($\displaystyle {\frac{{1}}{{2}}}$) =  2    f (2) =  1    g(3) = 3

The following plots are the graphs of f and g for -5≤t≤5.

dtbpFU2.0609in1.382in0ptf (x) or fpiecews1.wmf

For the function y = g(x) which is not continuous, you can plot with or without vertical connecting lines by using either the expression g(x) or the function name g to generate the plot. See Plotting Defined FunctionsDM6-4.tex#Plot piecewise-defined function for guidelines to plotting piecewise-defined functions.

itbpFU2.2675in1.5186in0ing(x)piecews4.wmf itbpFU2.0609in1.382in0ingpiecews2.wmf