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© 2014 Pacific Crest
105
M
ethodology
A
nalyzing
F
unctions
Scenario:
Analyze the function:
3
( )
1
f x
x
=
+
,
x
> 0.
Step
Explanation
Watch it Work!
1.
Convert to
function notation
Equations can sometimes represent
functions. Isolate the dependent
variable to create an expression
for function notation. Write the
transformed equation in function
notation form.
Already in function notation
2.
Determine the
function’s family
Families of functions have standard
features.
This is a rational function.
3.
Determine the
domain of the
function
If not specified, the domain is the set
of values for which the defining ex-
pression exists. Knowing the function’s
domain can help in determining the
appropriate graphing window.
The domain is specified as
x
> 0.
4.
Graph the
function
Many of the features of a function can
be determined from its graph.
12
–2
–2
8
y
x
f(x) =
3
1 +
x , x
> 0
5.
Asymptotes
A function’s horizontal and vertical
asymptotes can be determined from
its symbolic representation and can be
verified by its graph.
There is a no vertical
asymptote in this domain.
The horizontal asymptote
is the
x
-axis (
y
= 0).
6.
Determine the
range of the
function
The range of a function can be
determined from its graph and its
symbolic representation. A function’s
graph also gives information about its
maximum and minimum.
From the graph, the range is 0 <
x
< 3
or the
interval (0, 3) since
3 3
1 0
=
+
and as
x
gets large
the denominator
1 +
x
gets large and the defining
expression gets closer and closer to
zero. The function has no maximum
or minimum value.
7.
Determine
the function’s
intercepts
The intercepts of a function can be
determined from its graph or algebraic
representation.
There is no
x
-intercept or
y
-intercept.
2.5 Analyzing a Function