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Graphing a Function and Its Inverse Function
The graph of a function and its inverse are symmetric about the line
y
=
x
. Why do you think that this is so?
EXAMPLE 6
If
f
(
x
) = 3
x
−
2, find
f
−
1(
x
). Graph
f
and
f
−
1 on the same set of axes. Draw the line
y
=
x
as a dashed line for reference.
Solution
f
(
x
)
= 3
x
−
2
y
= 3
x
−
2
x
= 3
y
−
2
x
+ 2 = 3
y
begin numerator x plus 2 end numerator over 3 equals y
f to the
−
1 open parenthesis x close parenthesis power equals begin numerator x plus 2 end numerator over 3
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Now we graph each line.
We see that the graphs of
f
and
f
−
1 are symmetric about the line
y
=
x
. If we folded the graph paper along the line
y
=
x
, the graph of
f
would touch the graph of
f
−
1. Try it. Redraw the functions on a separate piece of graph paper. Fold
Student Practice 6
If
f open parenthesis x close parenthesis equals
−
1 over 4 x plus 1 comma
find
f
−
1
(
x
). Graph
f
and
f
−
1
on the same coordinate plane. Draw the line
y
=
x
as a dashed line for reference.
709710
11.4 Exercises Verbal and Writing Skills, Exercises 1–6
Complete the following:
1.
A one-to-one function is a function in which no ordered pairs _____________________________.
2.
If any horizontal line intersects the graph of a function more than once, the function ____________________.
3.
The graphs of a function
f
and its inverse
f
−
1
are symmetric about the line __________.
4.
Do all functions have inverse functions? Why or why not?
5.
Does the graph of a horizontal line represent a function? Why or why not? Does it represent a one-to-one function? Explain.
6.
Does the graph of a vertical line represent a function? Why or why not? Does it represent a one-to-one function? Explain.
Indicate whether each function is one-to-one.
7.
B
= {(0, 1), (1, 0), (10, 0)}
8.
A
= {(
−
6,
−
2), (6, 2), (3, 4)}
9.
10.
C
=
{(12, 3), (
−
6, 1), (6, 3)}
11.
E
= {(1, 2.8), (3, 6), (
−
1,
−
2.8), (2.8, 1)}
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12.
F
= {(6, 5), (
−
6,
−
5), (5, 6), (
−
5,
−
6)}
Indicate whether each graph represents a one-to-one function.
13.
14.
15.
16.
17.
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