Goal(s):
Define
piecewise functions
Evaluate piecewise functions for given
values
Determine if segments in a piecewise
function are increasing or decreasing
Determine if a given piecewise function is
continuous or discontinuous
Definition
Piecewise Function:
a function defined by multiple
sub functions, each
sub function applying to a
certain interval of the main
function's domain
What do they look like?
f(x) =
x2 + 1 , x  0
x–1, x0
You can EVALUATE piecewise
functions.
You can GRAPH piecewise functions.
Evaluating Piecewise Functions:
Evaluating piecewise functions is just
like evaluating functions that you are
already familiar with.
Let’s calculate f(2).
f(x) =
x2 + 1 , x  0
x–1, x0
You are being asked to find y when
x = 2. Since 2 is  0, you will only
substitute into the second part of the
function.
f(2) = 2 – 1 = 1
Let’s calculate f(-2).
f(x) =
x2 + 1 , x  0
x–1, x0
You are being asked to find y when
x = -2. Since -2 is  0, you will only
substitute into the first part of the
function.
f(-2) = (-2)2 + 1 = 5
Your turn:
f(x) =
2x + 1, x  0
2x + 2, x  0
Evaluate the following:
f(-2) = -3
?
f(5) = 12
?
f(0) = 2?
f(1) = 4?
One more:
f(x) =
3x - 2, x  -2
-x , -2  x  1
x2 – 7x, x  1
Evaluate the following:
f(-2) = 2?
?
f(3) = -12
f(-4) = -14
?
f(1) = -6?
Characteristics of functions
A function (or part of a function) is said
to be increasing if the y-values are
increasing as the x-values increase.
A function (or part of a function) is said
to be decreasing if the y-values are
decreasing as the x-values increase.
Characteristics of functions
A function is said to be continuous if
there are no breaks in the function (you
could draw it without picking up your
pencil)
A function is said to be discontinuous
if there are breaks in the domain or
range of the function (you would have
to pick up your pencil or erase a section
to draw it)