Evaluating Piecewise and Step Functions

1
Evaluating Piecewise and Step Functions

• Today, you will
• a) evaluate piecewise functions
• b) investigate and explain characteristics of a
  variety of piecewise functions including domain,
  range, vertex, axis of symmetry, extrema, points
  of discontinuity and rates of change

2
Evaluating Piecewise Functions

• Piecewise functions are functions defined by at
  least two equations, each of which applies to a
  different part of the domain
• There are several types of piecewise functions
  that can take on all different shapes and forms!
• A piecewise function looks like this

Domain restrictions
Equations
3
Evaluating Piecewise Functions

• When we evaluate piecewise functions, the most
  important thing to do is look at the individual
  domains for the functions and find which part of
  the piecewise function you will need to use.

For example, find a. g(-2) and b. g(2)
At g(-2) we would use the top function because -2
lt 1. So, g(-2) -2
At g(2) we would use the bottom function because
2 gt 1. So, g(2) 3(2) 1 5
4
Evaluating Piecewise Functions
Lets look at another example.
Which equation would we use to find g(-5)?
g(-2)? g(1)?
5
Step Functions
Step functions are special types of piecewise
functions that are defined by a constant value
over each part of its domain. Graphically, it
looks like a flight of stairs
An example of a step function
Graphically, the equation would look like this
6
Evaluating Step Functions

• To evaluate a step function, treat it just like
  any other piecewise function. Using the domain,
  identify which piece of the piecewise function
  you will need to use and identify the value.
• Two special kinds of step functions are called
  floor and ceiling functions. In ceiling
  functions, non-integers are rounded up to the
  nearest integer. In floor functions, all
  non-integers are rounded down.
• Example ceiling function you use 147 talking
  on the phone, but you are charged for 2 min.
• Example floor function you may be 14 years and
  8 months old, but you say you are 14 years old
  until your 15th birthday.

7
Characteristics of Piecewise Functions

• Piecewise functions, like all functions, have
  special characteristics. Some are familiar, some
  are new.

8
Domain and Range of Piecewise Functions

• Domain (x) the set of all input numbers - will
  not include points where the function(s) do not
  exist. The domain also controls which part of the
  piecewise function will be used over certain
  values of x.
• Range (y) the set of all outputs.

9
Points of Discontinuity

• With piecewise functions, we have what are called
  points of discontinuity. These are the points
  where the function either jumps up or down or
  where the function has a hole.
• For example, in a previous example Has a
  point of discontinuity at 1
• The step function also has points of
  discontinuity at 1, 2 and 3.

10
Axis of Symmetry

• In absolute value functions, there exists a
  vertical line that splits the equation in half.
  This axis of symmetry can be found by
  identifying the x-coordinate of the vertex (h,k),
  so the equation for the axis of symmetry would be
  x h.

For the equation the axis of
symmetry is located at x 1
11
Maxima and Minima

• Like all functions, piecewise functions have
  maxima and minima. These values will be a part
  of the range of the function

In this function, the minimum is at y 1 and the
maximum is infinity
In this function, the minimum is at y -2 and
the maximum is infinity
12
Intervals of Increase and Decrease

• By looking at the graph of a piecewise function,
  we can also see where its slope is increasing
  (interval of increase), where its slope is
  decreasing (interval of decrease) and where it is
  constant (slope is 0). We use the domain to
  define the interval.

This function is decreasing on the interval x lt
-2, is Increasing on the interval -2 lt x lt 1, and
constant over x gt 1