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

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:

Equations

Domain restrictions

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 < 1. So, g(-2) = -2

At g(2) we would use the bottom function because

2 > 1. So, g(2) = 3(2) – 1 = 5

Evaluating Piecewise FunctionsLets look at another example.

Which equation would we use to find; g(-5)? g(-2)? g(1)?

161)5(2)5()5( 2 g

11)2(2)2()2( 2 g41)1(2)1()1( 2 g

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:

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 1:47 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.

Characteristics of Piecewise Functions

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

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.

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.

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 xy

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

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 < -2, is Increasing on the interval -2 < x < 1, and constant over x > 1