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2
Analyze Square Root Functions
Previously you learned that an inverse relation
interchanges the
x
- and
y
-coordinates of the original relation. For the power function
f
(
x
)
=
x
2
, if the domain of
x
is restricted to nonnegative values, then the inverse of
f
is the
function
f
-
1
(
x
)
=
√
x
,
x
≥
0.
MUSIC
Refer to the application at the beginning of the lesson. The pitch, or
frequency, measured in hertz (Hz) of a certain string can be modeled by
f
(
T
)
=
1
_
1.28
√
T
_
0.0000708
, where
T
is tension in kilograms.
a.
Graph the function for tension in the domain {
T
|
0
≤
T
≤
10}.
Make a table of values for 0
≤
T
≤
10
and graph.
T y
(
T
)
0
0
1
92.8
2
131.3
3 160.8
4 185.7
5 207.6
T f
(
T
)
6 227.4
7 245.7
8 262.6
9 278.5
10 293.6
C07-013A-888482
Frequency (Hz)
160
200
80
120
40
0
240
280
320
360
400
Tension (kg)
2 3 1
8 9
4 5 6 7
10
T
f
(
T
)
b.
How much tension is needed for a pitch of over 200 Hz?
According to the graph and the table, more than 4.5 kilograms of tension
is needed for a pitch of more than 200 hertz.
Guided Practice
3.
MUSIC
The frequency of vibrations for a certain guitar string when it is plucked can
be determined by
F
=
200
√
T
, where
F
is the number of vibrations per second and
T
is the tension measured in pounds. Graph the function for 0
≤
T
≤
10. Then
determine the frequency for
T
=
3, 6, and 9 pounds.
Real-World Example 3
Use Graphs to Analyze Square Root Functions
Find the inverse of
f
(
x
)
=
x
2
,
x
≥
0. Graph
f
(
x
) and
f
-
1
(
x
) on the same coordinate plane.
f
(
x
)
=
x
2
Write the original function.
y
=
x
2
Replace
f
(
x
) with
y
.
x
=
y
2
Interchange
x
and
y
.
±
√
x
=
y
Take the square root of both sides.
Because the domain of
f
is restricted to nonnegative values
of
x
, the range of
f
–1
must also be restricted to nonnegative
values. So, the inverse of f is
f
–1
(
x
)
=
√
x
,
x
≥
0.
The graph of
f
–1
(
x
)
=
R
x
is a reflection of the graph of
f
(
x
)
=
x
2
,
x
≥
0, in the line
y
=
x
, shown as a dashed line on the graph.
Guided Practice
4.
Find the inverse of
f
(
x
)
=
x
2
+
1,
x
≥
0
and g
raph
f
(
x
) and
f
-
1
(
x
) on the same
coordinate plane.
C05_011A_903990
y
x
O
f
(
x
)
=
x
z
Example 4
Find the Inverse of Power Function
f
(
x
)
=
x
2
Problem-Solving Tip
Modeling
Making a
table is a good way to
organize ordered pairs in
order to see the general
behavior of a graph.
Problem-Solving Tip
Reasoning
Point out
that if the domain of
f
(
x
)
=
x
2
is restricted to
x
≤
0, its
inverse would be the
function
f
–1
(
x
)
=-
√
x
.
Real-World Link
On every string, the guitar
player has an option of
decreasing the length of the
string in about 24 different
ways. This will produce 24
different frequencies on
each string.
Source:
Guitar World
Cybermama/Getty Images
340
|
Lesson 5-4
|
Graphing Square Root Functions