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Calculus 1.3 Part 2
Functions and Graphs
Objectives
• Understand and Graph the Greatest Integer
Function.
• the graph of a function.
• Identify different types of transformations of
functions.
• Classify functions and recognize combinations
of functions
The Greatest Integer Function
• F(x)=[[x]] is called the greatest integer
function
• It is also denoted as:
f ( x) =  x  , and
is called the floor function.
• Our book uses f(x)=[[x]] as the greatest integer
function. (see page 92)
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The Greatest Integer Function
• To calculate. Say. “What is the greatest integer
that is less than or equal to x.”
•
•
•
•
Calculuate:
A) [3.5]
B) [-0.04]
C) [0]
The Greatest Integer Function
• To calculate. Say. “What is the greatest integer
that is less than or equal to x.”
•
•
•
•
Calculuate:
A) [3.5]
B) [-0.04]
C) [0]
Graph and State the Domain and Range
• G(x)=[x].use open and closed circles.
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Graph
G(x)=[x+1].use open and closed circles.
Look for a pattern!
The Graph
A Function Defined by More than one
Equation
• The piecewise function:
1 − x, if x < 1
f ( x) = 
 x − 1, if x ≥ 1
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Transformations of Functions
• Consider the parent graph to be:
y = f ( x)
y = f ( x − c) + b
Would translate the parent c
units right and b units up
Transformations of Functions
•
The following would translate the function c units left and b units down.
y = f ( x + c) − b
Reflections are obtained by y=–f(x)
which reflects over the x-axis and
y=f(-x) would reflect over the y-axis.
Vertical stretch or shrink by a factor of a is obtained by y =
af(x)
Horizontal stretch or shrink by a factor of 1/a is obtained
by y = f(ax)
Composition of Functions
• Let f and g be functions. The functions
given by
( f g )( x) = f ( g ( x))
Is called the composite of f with g. The domain of f(g(x))
is the set of all x in the domain of g such that g(x) is in the
domain of f
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Composition of Functions
• Let f and g be functions. The functions
given by
( f g )( x) = f ( g ( x))
Is called the composite of f with g. The domain of f(g(x))
is the set of all x in the domain of g such that g(x) is in the
domain of f
Practice Problems
• P27(24, 28, 29,30,31-37 odd, 41-44 all,45-72
multiples of 3, 73-77 all)
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