Math Problem Statement

反函数

反函数

Solution

The given equation is:

y=ex+12y = e^{x+1} - 2

To find the inverse function of this equation, we need to follow these steps:

Step 1: Express xx in terms of yy.

We start by solving for xx.

  1. Add 2 to both sides: y+2=ex+1y + 2 = e^{x+1}

  2. Take the natural logarithm (ln) of both sides to get rid of the exponential: ln(y+2)=x+1\ln(y + 2) = x + 1

  3. Subtract 1 from both sides to isolate xx: x=ln(y+2)1x = \ln(y + 2) - 1

Step 2: Express the inverse function.

Now, we can rewrite the equation as: x=ln(y+2)1x = \ln(y + 2) - 1

Swapping xx and yy to express the inverse function: y=ln(x+2)1y = \ln(x + 2) - 1

Thus, the inverse function is: f1(x)=ln(x+2)1f^{-1}(x) = \ln(x + 2) - 1

Summary:

The inverse of the given function y=ex+12y = e^{x+1} - 2 is f1(x)=ln(x+2)1f^{-1}(x) = \ln(x + 2) - 1.

Would you like more details or have any questions?

Related Questions:

  1. How can you verify if a function and its inverse are correct?
  2. What is the domain and range of the original function y=ex+12y = e^{x+1} - 2?
  3. What is the domain and range of the inverse function f1(x)f^{-1}(x)?
  4. How do natural logarithms help in solving equations involving exponentials?
  5. What are the key properties of exponential and logarithmic functions?

Tip:

To confirm that two functions are inverses, you can compose them: if f(f1(x))=xf(f^{-1}(x)) = x and f1(f(x))=xf^{-1}(f(x)) = x, 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