To find the inverse function \( f^{-1}(x) \) of the given function \( f(x+1) = 2x \), we need to express \( x \) in terms of \( y \), where \( y = f(x+1) \).

### Step 1: Start with the original function
The function is given as:
\[
f(x+1) = 2x
\]
Let's set \( y = f(x+1) \), so:
\[
y = 2x
\]

### Step 2: Solve for \( x \)
To find the inverse, solve for \( x \) in terms of \( y \):
\[
x = \frac{y}{2}
\]

### Step 3: Express the inverse function
Remember, \( y = f(x+1) \), and we found that \( x = \frac{y}{2} \). Since \( y = f(x+1) \), replace \( y \) with \( x \) in the inverse function:
\[
f^{-1}(x) = \frac{x}{2} - 1
\]

### Final Answer
The inverse function \( f^{-1}(x) \) is:
\[
\boxed{f^{-1}(x) = \frac{x}{2} - 1}
\]

This is the function that, when applied to the output of \( f(x+1) \), returns the original input.

Do you have any questions or would you like further details?

Here are some related questions to explore:

1. How can you verify if a function and its inverse are correct?
2. What are the conditions under which a function has an inverse?
3. How do you find the inverse of a more complex function, such as a quadratic function?
4. What is the graphical relationship between a function and its inverse?
5. Can you find the inverse of the function \( g(3x) = x + 6 \)?

**Tip:** To verify the correctness of an inverse function, compose the original function with its inverse (i.e., \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \)). Both compositions should return \( x \).

Learn how to find the inverse function of f(x+1) = 2x step-by-step. This problem involves understanding functions and inverse functions. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation