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por que si lo despejo de otra forma me da -x-1/x-1 y otra forma me da x+1/-x+1

To solve the problem and clarify the confusion, let's go through the steps carefully. The given function is:

\[
g(x) = \frac{x - 1}{x + 1}
\]

We need to find the inverse function \( g^{-1}(x) \) and evaluate \( g^{-1}(-1) \).

### Step 1: Find the inverse function \( g^{-1}(x) \)
To find the inverse, we swap \( g(x) \) with \( y \) and \( x \):

\[
y = \frac{x - 1}{x + 1}
\]

Now, solve for \( x \) in terms of \( y \):

1. Multiply both sides by \( x + 1 \) to eliminate the denominator:

\[
y(x + 1) = x - 1
\]

2. Distribute \( y \) on the left side:

\[
yx + y = x - 1
\]

3. Get all terms involving \( x \) on one side and the rest on the other:

\[
yx - x = -1 - y
\]

4. Factor out \( x \) from the left side:

\[
x(y - 1) = -1 - y
\]

5. Solve for \( x \):

\[
x = \frac{-1 - y}{y - 1}
\]

This expression represents \( g^{-1}(y) \), so:

\[
g^{-1}(x) = \frac{-1 - x}{x - 1}
\]

### Step 2: Evaluate \( g^{-1}(-1) \)

Now, substitute \( x = -1 \) into the inverse function:

\[
g^{-1}(-1) = \frac{-1 - (-1)}{-1 - 1} = \frac{0}{-2} = 0
\]

So, \( g^{-1}(-1) = 0 \).

### Addressing the Confusion

If you obtained different expressions such as:

\[
\frac{-x - 1}{x - 1} \quad \text{or} \quad \frac{x + 1}{-x + 1}
\]

These might result from different algebraic manipulations or factoring methods. However, upon simplification, they should all represent the same function as long as the operations are correctly followed. Specifically:

1. \( \frac{-x - 1}{x - 1} \) simplifies by factoring out \(-1\) from the numerator, leading to the same function.
2. \( \frac{x + 1}{-x + 1} \) is the same as \( \frac{-(x + 1)}{x - 1} \), which also matches the derived inverse.

### Summary

The correct inverse is \( g^{-1}(x) = \frac{-1 - x}{x - 1} \), and \( g^{-1}(-1) = 0 \).

Do you have any questions or need more details?

### Related Questions
1. How do you verify that two expressions are equivalent algebraically?
2. What are the key steps to finding an inverse function?
3. How do you handle fractions involving polynomials?
4. Can the inverse of a function be calculated graphically?
5. What are common errors when finding inverse functions?

### Tip
Always check your work by substituting the inverse function back into the original to ensure they undo each other (i.e., \( g(g^{-1}(x)) = x \)).

Si g(x) = (x - 1) / (x + 1), encuentre g^(-1)(-1).

Learn how to find the inverse function of g(x) = (x - 1) / (x + 1) and evaluate g^(-1)(-1) with a clear, step-by-step solution. This problem involves key algebraic concepts and is suitable for students in Grades 10-12.

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