Special Piecewise Functions Solutions

• The following special piecewise functions do not require that you note the domain intervals:
  •    absolute value    functions
  •     step    functions
• Because the domain and range are not listed for these functions, you must remember:
  • what the    notation    for the function means,
  • as well as the    domain and range    .
    

Absolute Value Function

Step Functions

Function	Greatest Integer	Least Integer
Also Known As	floor function	ceiling function
y-value	rounded down to the lower of the two integers that it is between	rounded up to the higher of the two integers that it is between
Closed Point Value	the lower value is an included point	the higher value is an included point
Equation	$f\left(x\right)=\lfloor x\rfloor$	$h\left(x\right)=\lceil x\rceil$
Parent Graph
Domain and Range	domain: $\left\{x|x\in \mathrm{ℝ}\right\}$ range: $\left\{y|y\in \mathrm{\mathbb{Z}}\right\}$	domain: $\left\{x|x\in \mathrm{ℝ}\right\}$ range: $\left\{y|y\in \mathrm{\mathbb{Z}}\right\}$

Note

Step functions pass the VLT because when the points align vertically, one is open (excluded) and one is closed (included), so there is no overlap.

Example 6

Finish the graph of the step function. Name the domain and range for the function. 

Describe how the graph is translated as compared to the parent graph. 

The y-intercept is $\underline{ (0, 3) }$ rather than (0, 0) which means the function is    translated up 3 spaces   .

Example 7

Graph.

Note

When looking at an equation, it is best practice to think about how it relates to the parent function and how it is translated.