Combining
Functions
Functions are added, subtracted, multiplied, divided in the usual way.
There is really nothing new here, except for the notation.
Given functions 
and 
,
addition
of function
difference of functions
product
of functions
quotient of functions
See Example 1, pages 242 - 243.
Composition
This is new.
Given functions
and ,
their composition, or composite, is defined by
The domain of 
is the set of all real numbers 
in the domain of
such that 
is in the domain of .
Example
Let 
and 
.
To avoid confusion with the variable 
,
rewrite the functions using boxes.
Wherever appears,
replace it with an empty box,
the box here is called the -box
the box here is called the -box
says
that we put inside
the -box,
Similarly,
says that we put
inside the -box
See Examples 2 - 4, pages 244 - 246.
Vertical Translations
To graph 
shift the graph of 
up 
units if 
down
units if
.
Horizontal Translations
To graph 
shift the graph of
left 
units if 
right
units
if 
Reflections
To graph 
reflect the graph of
across the x-axis
Vertical Stretchings and Shrinkings
To graph 
if 
vertically stretch the graph of
by
multiplying each ordinate value of the graph
of
by 
if 
vertically shrink the graph of
by multiplying each ordinate value of the graph
of
by
See Figure 9, page 251 for more images illustrating
the above transformations of graphs.
See Examples 5 - 8, pages 247 - 252.
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Functions