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Polynomial and rational functions
Inverses and radical functions
Finding inverses of rational functions
As with finding inverses of quadratic functions, it is sometimes desirable to find the
inverse of a rational function , particularly of rational functions that are the ratio of linear functions, such as in concentration applications.
Finding the inverse of a rational function
The function
C
=
20
+
0.4
n
100
+
n
represents the concentration
C
of an acid solution after
n
mL of 40% solution has been added to 100 mL of a 20% solution. First, find the inverse of the function; that is, find an expression for
n
in terms of
C
.
Then use your result to determine how much of the 40% solution should be added so that the final mixture is a 35% solution.
We first want the inverse of the function. We will solve for
n
in terms of
C
.
C
=
20
+
0.4
n
100
+
n
C
(
100
+
n
)
=
20
+
0.4
n
100
C
+
C
n
=
20
+
0.4
n
100
C
−
20
=
0.4
n
−
C
n
100
C
−
20
=
(
0.4
−
C
)
n
n
=
100
C
−
20
0.4
−
C
Now evaluate this function for
C
=
0.35
(
35
%
)
.
n
=
100
(
0.35
)
−
20
0.4
−
0.35
=
15
0.05
=
300
We can conclude that 300 mL of the 40% solution should be added.
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Key concepts
The inverse of a quadratic function is a square root function.
If
f
−
1
is the inverse of a function
f
,
then
f
is the inverse of the function
f
−
1
.
See
[link] .
While it is not possible to find an inverse of most polynomial functions, some basic polynomials are invertible. See
[link] .
To find the inverse of certain functions, we must restrict the function to a domain on which it will be one-to-one. See
[link] and
[link] .
When finding the inverse of a radical function, we need a restriction on the domain of the answer. See
[link] and
[link] .
Inverse and radical and functions can be used to solve application problems. See
[link] and
[link] .
Section exercises
Verbal
Algebraic
For the following exercises, find the inverse of the function on the given domain.
For the following exercises, find the inverse of the functions.
For the following exercises, find the inverse of the functions.
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Source:
OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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