How to solve inverse functions for multiple power of x?
Answers
Step-by-step explanation:
will go over three examples in this tutorial showing how to determine algebraically the inverse of an exponential function. But before you take a look at the worked examples, I suggest that you review the suggested steps below first in order to have a good grasp of the general procedure.
Steps to Find the Inverse of an Exponential Function
STEP 1: Change f (x) to y.
f (x) → y
STEP 2: Interchange x and y in the equation.
x → y
y → x
STEP 3: Isolate the exponential expression on one side (left or right) of the equation.
The exponential expression shown below is a generic form where “b” is the base, while “N” is the exponent.

STEP 4: Eliminate the base “b” of the exponential expression by taking the logarithms of both sides of the equation.
To make the simplification much easier, take the logarithm of both sides using the base of the exponential expression itself.
Using the log rule,


STEP 5: Solve the exponential equation for “y” to get the inverse. Finally, replace y with the inverse notation f -1(x) to write the final answer.
Rewrite y as f -1 (x)
Let’s apply the suggested steps above to solve some problems.
Examples of How to Find the Inverse of an Exponential Function
Example 1: Find the inverse of the exponential function below.

This should be an easy problem because the exponential expression on the right side of the equation is already isolated for us.
Start by replacing the function notation f (x) by y.

Next step is to switch the variables x and y in the equation.

Since the exponential expression is by itself on one side of the equation, we can now take the logarithms of both sides. When we get the logarithms of both sides, we will use the base of “2” because this is the base of the given exponential expression.

Apply the Log of Exponent Rule, , in the simplification. The rule states that the logarithm of an exponential number where its base is the same as the base of the log is equal to the exponent.

We are almost done! Solve for yby adding both sides by 5, and then divide the equation by the coefficient of y which is 3. Don’t forget to change y to f -1(x) to mean that we have the inverse function.

If we graph the original exponential function and its