f(x)=8÷x-3. Find f^-1(x)
Answers
Step-by-step explanation:
For the sake of easier notation, we shall say that
f
is a function with inverse function
g
, that is,
f
−
1
(
x
)
=
g
(
x
)
.
According to the definition of inverse functions,
f
(
g
(
x
)
)
=
x
Differentiating through the chain rule gives
f
'
(
g
(
x
)
)
g
'
(
x
)
=
1
Solving for the derivative of the inverse gives
g
'
(
x
)
=
1
f
'
(
g
(
x
)
)
So, we want to find
g
'
(
2
)
.
g
'
(
2
)
=
1
f
'
(
g
(
2
)
)
We want to first find
g
(
2
)
, however, we cannot write an expression for
g
(
x
)
. What we have to remember is that the domain of the mother function is the range of its inverse function, and vice versa.
Note that if
f
(
x
)
=
2
, then
x
=
g
(
2
)
. This means we should let
f
(
x
)
=
2
then solve for
x
, which is equal to
g
(
2
)
.
f
(
x
)
=
2
⇒
x
5
+
3
x
−
2
=
2
Continuing to solve yields
x
5
+
3
x
−
4
=
0
This may look impossible, but notice that the sum of the coefficients of each term is
0
, that is,
1
+
3
−
4
=
0
. This means that
x
=
1
is a solution.
Dividing, we see that
(
x
−
1
)
(
x
4
+
x
3
+
x
2
+
x
+
4
)
=
0
Note that
x
4
+
x
3
+
x
2
+
x
+
4
>
0
for all values of
x
, so it has no real roots. Thus,
f
(
x
)
=
2
when
x
=
1
, i.e.,
f
(
1
)
=
2
.
This means that
g
(
2
)
=
1
.
Returning to what we had earlier, we see that
g
'
(
2
)
=
1
f
'
(
g
(
2
)
)
=
1
f
'
(
1
)
Find this by taking the derivative of
f
.
f
(
x
)
=
x
5
+
3
x
−
2
⇒
f
'
(
x
)
=
5
x
4
+
3
Thus
f
'
(
1
)
=
5
+
3
=
8
.
g
'
(
2
)
=
1
8
With your notation,
[
f
−
1
]
'
(
2
)
=
1
8