Math Problem Statement
f^-1(x) = e^2x+5
Solution
The given function is:
This represents the inverse function of some original function . To find , we need to follow these steps:
Step 1: Replace with
Let .
Step 2: Swap and
We switch and to find the original function:
Step 3: Solve for
To solve for , follow these steps:
- Take the natural logarithm on both sides to eliminate the exponential function:
- Isolate :
Thus, the original function is:
Summary:
The original function is:
Would you like further details or have any questions?
Here are 5 related questions for further exploration:
- How do you find the derivative of ?
- What are the domain and range of ?
- How do you compute the inverse of logarithmic functions in general?
- How can we verify that and ?
- How does the natural logarithm function relate to exponential functions?
Tip: When working with logarithmic and exponential equations, taking logarithms on both sides is a common strategy to simplify the expression.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Inverse Functions
Exponential Functions
Logarithmic Functions
Formulas
Inverse of an exponential function
Natural logarithm: ln(x)
Basic algebraic manipulation
Theorems
Inverse Function Theorem
Properties of Logarithms and Exponentials
Suitable Grade Level
Grades 10-12