2.7 Piecewise-Defined Functions

Some functions are defined in pieces by different formulas in different parts of the domain. We saw two of such functions in Example 1 of the previous section. We call such functions piecewise-defined functions.

Absolute Value Function

An important example of a piecewise defined function is the absolute value function $f(x)=|x|$, which can also be written as:

$$f(x)=\{\begin{array}{cc}x& \text{if }x\ge 0\\ -x& \text{if }x<0\end{array}$$

The graph of this function is shown in Fig 1.

Sign Function

Another important piecewise defined function is the sign (or signum) function. The sign function, often denoted by $\mathrm{s}\mathrm{g}\mathrm{n}$ or $\text{sign}$, extracts the sign of a real number $x$ (see Fig 2.)

$$\mathrm{s}\mathrm{g}\mathrm{n}(x)=\{\begin{array}{cc}1& \text{if }x>0\\ 0& \text{if }x=0\\ -1& \text{if }x<0\end{array}$$

Greatest Integer Function or Floor Function

A function that takes a real number $x$ as input and returns the largest integer less than or equal to $x$ is called the greatest integer function or the  floor function. It is denoted $\lfloor x\rfloor$. In fact, this function rounds down a real number to the nearest integer. For example

$$\lfloor 7\rfloor =7,\lfloor 1.34\rfloor =1,\lfloor 3.99\rfloor =3,$$ $$\lfloor -7\rfloor =-7,\lfloor -1.34\rfloor =-2,\lfloor -3.99\rfloor =-4,$$ $$\lfloor \sqrt{2}\rfloor =1,\lfloor 0.12\rfloor =0$$

$$\lfloor -\sqrt{2}\rfloor =-2,\lfloor -0.12\rfloor =-1$$

• The greatest integer function is a function from $\mathbb{R}$ to $\mathbb{Z}$ (set of all integers)
  $$\mathbb{R}\to \mathbb{Z}$$
• Using mathematical notations, we can write
  $$\lfloor x\rfloor =max\{m\in \mathbb{Z}|m\le x\}.$$
•  Other symbols for the greatest integer function are $[x]$ and $[[x]]$.
  

  

  
  The graph of $y=\lfloor x\rfloor$ is shown in Figure 3.

Solution

Because $2\in (-1,2]$, to evaluate $f(2)$ we use the middle formula:
$$f(2)=x-1{|}_{x=2}=2-1=1.$$ Also
$$0\in (-1,2]\Rightarrow f(0)=x-1{|}_{x=0}=0-1=-1.$$ To evaluate $f(-1)$, we note that $-1\in (-\mathrm{\infty },-1]$. Therefore, we should use the bottom formula:

$$f(-1)=-2x+1{|}_{x=-1}=-2\times (-1)+1=3.$$ Fig 4 represents the graph of $f$.