To find the inverse of the function \( y = \frac{1 - x}{1 + x} \), follow these steps:

### Step 1: Replace \( y \) with \( f(x) \)
We start by writing the function as:
\[
y = \frac{1 - x}{1 + x}
\]

### Step 2: Switch \( x \) and \( y \)
For the inverse, we swap \( x \) and \( y \):
\[
x = \frac{1 - y}{1 + y}
\]

### Step 3: Solve for \( y \)
Now we solve for \( y \) in terms of \( x \).

1. Multiply both sides by \( 1 + y \) to get rid of the denominator:
   \[
   x(1 + y) = 1 - y
   \]

2. Expand both sides:
   \[
   x + xy = 1 - y
   \]

3. Group all terms with \( y \) on one side:
   \[
   xy + y = 1 - x
   \]

4. Factor out \( y \):
   \[
   y(x + 1) = 1 - x
   \]

5. Solve for \( y \):
   \[
   y = \frac{1 - x}{x + 1}
   \]

### Final Answer:
Thus, the inverse function is:
\[
f^{-1}(x) = \frac{1 - x}{x + 1}
\]

It turns out that the inverse of the given function is the same as the original function, meaning the function is **self-inverse**.

Would you like further clarification or details on any steps?

### Related Questions:
1. What are the conditions for a function to have an inverse?
2. How can you verify if a function is its own inverse?
3. What is a graphical interpretation of a function and its inverse?
4. How do you find the domain and range of this function and its inverse?
5. Can you provide other examples of self-inverse functions?

### Tip:
A function that is its own inverse is called an **involution**. This property can be useful in simplifying calculations.

What is the inverse of y = (1 - x) / (1 + x)?

Find the inverse of the function y = (1 - x) / (1 + x) with a clear, step-by-step solution. Explore key algebraic concepts and learn how to solve inverse functions. This problem is 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