# Composition of Functions

**Composition** of the function f with the function g,
given by , is defined by:

Domain of consists of all x in domain of g such that is in the domain of f.

Example. :If and , find:

Example: Use the graphs of f and g to evaluate each expression.

Example: In the example above, for what value (s) of x is ?

**Horizontal Compression and Elongation of Graphs .** If
,

a) For any constant , the graph of
is the same as that of
with a

change in the horizontal scale.

1) If horizontal compression

2) If horizontal elongation

b) The graph of is a **reflection** about
the y-axis of the graph of .

1) For horizontal compression and
reflection.

2) For horizontal elongation and reflection.

Example: Graph the following functions:

Example: If , , and , find:

Example: Use the graph of the function given below to
sketch the graph of the indicated

functions.

Example: Write the function
as the composition of three functions.

Applied Example: The number of bacteria in a refrigerated food is given by

where t is the Celsius temperature of the
food. when

the food is removed from refrigeration , the temperature is given by

where t is the time in hours.

a) What does the composite function
represent?

b) How many bacteria are in the food when t=2 hours?

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