Math Problem Statement
反函数
Solution
The given equation is:
To find the inverse function of this equation, we need to follow these steps:
Step 1: Express in terms of .
We start by solving for .
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Add 2 to both sides:
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Take the natural logarithm (ln) of both sides to get rid of the exponential:
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Subtract 1 from both sides to isolate :
Step 2: Express the inverse function.
Now, we can rewrite the equation as:
Swapping and to express the inverse function:
Thus, the inverse function is:
Summary:
The inverse of the given function is .
Would you like more details or have any questions?
Related Questions:
- How can you verify if a function and its inverse are correct?
- What is the domain and range of the original function ?
- What is the domain and range of the inverse function ?
- How do natural logarithms help in solving equations involving exponentials?
- What are the key properties of exponential and logarithmic functions?
Tip:
To confirm that two functions are inverses, you can compose them: if and , they are indeed inverses.
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Math Problem Analysis
Mathematical Concepts
Inverse Functions
Exponential Functions
Logarithmic Functions
Formulas
Inverse Function: f^{-1}(x)
Exponential Function: e^{x}
Logarithmic Function: \ln(x)
Theorems
Properties of Exponential and Logarithmic Functions
Suitable Grade Level
Grades 10-12