Mathematics · Ch 15 — Functions

Piecewise Defined Functions

15.2.3

Piecewise Defined Functions

Piecewise defined functions. A function defined by two or more separate equations, each applying only on its own specified part of the domain, is called a piecewise defined function.

Example: $f(x)=\begin{cases}4-x, & x\ge1\\ x+1, & x<1\end{cases}$ . Here $f(3)=4-3=1$ (since $3>1$ uses the first piece), $f(-2)=-2+1=-1$ (since $-2<1$ uses the second piece), and $f(1)=4-1=3$ (since $1\ge1$ uses the first piece, being the boundary point). …

Figure 1Fig. 6.37 — graph of a two-piece function with an open and closed dot at the join

What this figure shows. A graph made of two straight-line pieces meeting near x=1: for x greater than or equal to 1 the line y=4-x is drawn (shown as a solid filled-in/black dot at the point (1,3), since x=1 belongs to this piece), and for x less than 1 the line y=x+1 is drawn (shown as an open/hollow white dot at the point (1,2), since x=1 does NOT belong to this piece, even though the formula would give that value there). Illustrates the standard convention of using a filled dot for an included endpoint and a hollow/open dot for an excluded one at the seam between two pieces of a pi …

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