Special Functions
Step
Constant
Identity
Absolute Value
Piece Wise
Step Function
or Greatest Integer Function
written [[x]] in the book
The step function looks like stairs!
The symbol [[x]] means
• the greatest integer less than or equal to x.
• Therefore, [[7.5]]= 7
[[-1.5]]= -2 because -1 > -1.5
[[1.5]]= 1
The most common real life
example is postage.
A one ounce letter costs 34 cents. If it weights 1.5
ounces it still costs 34 cents! The cost changes with each
ounce. Thus forming the steps.
Each step has a closed circle on
one side of the step and an open
circle on the other.
What does the closed circle represent?
What does the open circle represent?
Example 1 p.89 and 90.
The Constant Function
The constant function is a horizontal line.
The equation for a horizontal line is?
y= any number
It has no slope or zero slope!
The Identity Function
The identity function is y=x,
where the slope is 1 and the
y-intercept is 0.
Absolute Value Function
Recall that the absolute value of a positive or a negative number
is always positive. Whether we substitute -1 or 1 for x we will
still have y=1. This is true for all real numbers substituted for x.
The Absolute Value Function
• Example p.91 in the book
• Absolute Value Dynamic Worksheet
The absolute value function can be written as:
f(x) = {-x if x < 0, x if x > 0}
A function using two or more expressions is called a
piecewise function.
Piecewise Function
A piecewise function is a combination of more
than one function graphed according to limits
placed on each function.
http://www.math.uky.edu/~pkoester/teaching/Fall_2009/Math123/GeoGebra/Contin
uous1.html
Piecewise Function
• Example p.92
• Piecewise Applet
• Piecewise Functions
Graphing Special Functions
• When graphing these functions by hand it is
easiest to create a table of values.
• Then recall the qualities of the function to
complete the graph.
• Do the problems on page 94 #24, 26, 30, 32,
34, 38, and 40. Then work on the
worksheets.