If the ordered pairs of a relation
R are reversed, then the new set of ordered pairs is called the
inverse relation of the original relation.
Example 1: If
R = {(1,2), (3,8), (5,6)}, find the inverse relation of
R. (The inverse relation of
R is written
R−1).
Notice that the domain of
R−1 is the range of
R, and the range of
R−1 is the domain of
R. If a relation and its inverse are graphed, they will be symmetrical about the line
y =
x.
Example 2: Graph
R and
R−1 from Example 1 along with the line
y = x on the same set of coordinate axes.
The answer is shown in Figure
1 .
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Figure 1
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Symmetrical sets of points.
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If this graph were “folded over” the line
y = x, the set of points called
R would coincide with the set of points called
R−1, making the two sets symmetrical about the line
y = x.
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Identity function. The function
y = x, or
f (
x) =
x, is called the
identity function, since for each replacement of
x, the result is identical to
x.
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Inverse function. Two functions,
f and
g, are inverses of each other when the composition
f [
g(
x)] and
g[
f (
x)] are both the identity function. That is,
f [
g(
x)] =
g[
f (
x)] =
x.
Example 3: If
f (
x) = 4
x − 5, find
f−1(
x).
To find
f−1(
x), simply reverse the
x and
y variables and solve for
y.
For any ordered pair that makes
f(
x) = 4
x − 5 true, the reverse ordered pair will make
true.
To show that
f (
x) and
f−1(
x) are truly inverses, show that their compositions both equal the identity function.
Since
f [
f−1(
x)] =
f−1[
f(
x)] =
x, then
f(
x) and
f−1(
x) are inverses of each other.
Example 4: Graph
f (
x) and
f−1(
x) from Example 13 together with the identity function on the same set of coordinate axes. The answer is shown in Figure
2 .
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Figure 2
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Symmetrical graphs.
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Notice that if the graph were “folded over” the identity function, the graphs of
f (
x) and
f−1(
x) would coincide.
Example 5: If
f (
x) =
x2, find
f−1(
x).
There are two relations for
f−1(
x),
In order for both
f (
x) and
f−1(
x) to be functions, a restriction needs to be made on the domain of
f (
x) so only one relation appears as
f−1(
x). If the domain of
f (
x) is restricted to {
x|
x ≥ 0}, the
is the only answer for
f−1(
x).
Example 6: Graph
f (
x) −
x2 together with
, and the identity function
f (
x) =
x all on the same set of coordinate axes.
To graph
f (
x) =
x2, find several ordered pairs that make the sentence
y = x2 true. To graph
, simply take the reverse of the ordered pairs found for
f (
x) =
x2. The graph is as shown in Figure
3 .
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Figure 3
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f−1(
x) is not a function.
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Notice that
f (
x) =
x2 is a function but that
is not a function. The reason is that
does not pass the vertical line test. Also notice that
f (
x) and
f−1(
x) will coincide when the graph is “folded over” the identity function. Thus, the two relations are inverses of each other.
Example 7: Graph
f(
x) =
x2 with the restricted domain {
x|x ≥ 0} together with
and the identity function on the same set of coordinate axes. The answer is shown in Figure
4 .
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Figure 4
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Solution to Example 17.
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Notice that
f (
x) and
f−1(
x) are now both functions, and they are symmetrical with respect to
f (
x) =
x. To show that
f (
x) =
x2 and
are inverse functions, show that their compositions each produce the identity function.
Linear Sentences in One Variable
Relations and Functions
