How to Find an Inverse Function
Recall that the inverse function of 
is defined as that function 
such that 
and 
. Therefore, we can obtain it from .
Similarly, the inverse function of is usually denoted as 
(note that the 
in the previous expression does not refer to a negative exponent, but only indicates that it is the inverse function).
Note: In general, for a function to have an inverse function, it is necessary that the function be one-to-one (or bijective). When this is not fulfilled, it is necessary to restrict the domain.
Recall that a one-to-one function is one that assigns a different value in the range to each element of the domain. That is, if 
then 
.
Method to Find the Inverse Function
1 Replace with 
.
2 Solve for the variable 
. From this we obtain an expression of the form 
3 In 
, replace the 's with .
4 Finally, change the on the left side to .
Example: Consider the function 
. We will follow the procedure to find the inverse function:
1 We replace with : 
.
2 We solve for :
where 
3 We interchange the 's with :
4 Then we change the on the left side to :
Finally, we verify that the function is indeed the inverse:
from which we can observe that 
is satisfied.
Proposed Exercises
Find the inverse function of the following linear function:
We will find the function without listing the steps. We have 
, where we replace with :
Then, we solve for :
Finally, we replace with and with :
which is the inverse function.
Find the inverse function of the following function:
First, we replace with :
Then we solve for :
That is:
Finally, we replace with and with :
which is the inverse function.
Find the inverse function of the following function (you do not need to simplify):
We start by replacing with :
Then we solve for . For this, we first multiply by :
Then we move the terms to one side of the equation and the remaining terms to the other:
Finally, we divide by 
:
Therefore, the inverse function is:
Calculate the inverse function of the following quadratic function:
Note that 
is not a one-to-one function (for example, 
). Therefore, it does not have an inverse function on the entire domain.
However, if we consider the domain as the interval 
, then the function will be one-to-one. In this case, the inverse is obtained as follows:
Solving for (and using the fact that 
in the restricted domain):
Therefore, in this case the inverse function is:
On the other hand, if we restricted the domain to 
, then the inverse function is obtained as follows:
Then we solve for (which satisfies 
):
Consequently, the inverse function is:
This means that 
is the inverse of only when the domain is the non-negative real numbers . If the domain is all real numbers, the function has no inverse.
Find the inverse function of the following function:
We start by replacing with :
Then we solve for :
Therefore, the inverse function is:
Find the inverse function of the following function:
We start by replacing with :
Then, recall that the natural logarithm satisfies:
Thus, we apply the natural logarithm to both sides of the equation:
in this way:
Therefore, the inverse function is:
Find the inverse function of the following function:
Radical functions are indeed one-to-one, therefore, they do have an inverse function:
where we know that 
. Then we square both sides of the equation:
Therefore, the inverse function is:
where (as we replace with , then at the end is the one that satisfies ).
In other words, for 
to be the inverse function of 
, it must be true that has as its domain only .
Find the inverse function of:
We know that the cube root function is one-to-one, has all real numbers as its domain, and all real numbers as its range. Therefore, it will have an inverse whose domain is all real numbers:
We cube both sides:
that is:
Therefore, the inverse function is:
Find the inverse function of:
Also, verify that
First, we find the inverse function. For this, we replace with :
Then, we solve for :
that is,
We already have solved. However, we simplify a bit:
Therefore, the inverse is:
Now, we verify what we were asked:
a)
First, we verify that 
. For this, we replace with its value:
Then, we evaluate with the given argument:
We simplify:
that is,
So the first relation is satisfied.
b)
Now we will verify that 
. First, we replace with its expression:
Then we evaluate :
that is,
therefore, the second relation is also satisfied.
Calculate the inverse of the following function:
and verify that .
We start by calculating the inverse, so we replace with :
Then, we solve for ; so we multiply by 
:
Then, we move the terms with toward the left side of the equation, and the remaining terms to the right side:
Therefore:
That is, the inverse function is:
Now we will verify that is satisfied. First, we replace the expression of :
Now we evaluate the inverse:
We simplify:
Then:
Therefore, the relation is satisfied.
Calculate the inverse of the function:
We start by replacing with , then solve for :
That is,
Calculate the inverse of the function: 
in the appropriate domain.
First, note that this is not an injective function. In fact, 
. Thus, we seek the inverse in the domain 
. We start by replacing with , then solve for :
That is,
Calculate the inverse of the function: 
We start by replacing with , then solve for :
That is,
Calculate the inverse of the function: 
We start by replacing with , then solve for :
That is,
Calculate the inverse of the function:
We start by replacing with , then solve for :
That is,
Summarize with AI:
Did you like this article? Rate it!
Loading...
