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To determine which functions are the same as their inverse functions, we need to examine each function and verify whether the function is self-inverse.

A function \( f(x) \) is the same as its inverse if:

\[ f(f(x)) = x \]

Let's go through each function and check this condition.

### 1. \( f(x) = \frac{x - 1}{x + 5} \)
Find the inverse by solving \( y = \frac{x - 1}{x + 5} \) for \( x \):

\[
y(x + 5) = x - 1
\]
\[
yx + 5y = x - 1
\]
\[
yx - x = -1 - 5y
\]
\[
x(y - 1) = -1 - 5y
\]
\[
x = \frac{-1 - 5y}{y - 1}
\]
Thus, the inverse of \( f(x) \) is \( f^{-1}(x) = \frac{-1 - 5x}{x - 1} \), which is not equal to \( f(x) \). Therefore, \( f(x) \) is not self-inverse.

### 2. \( g(x) = \frac{x - 2}{x - 1} \)
Find the inverse by solving \( y = \frac{x - 2}{x - 1} \) for \( x \):

\[
y(x - 1) = x - 2
\]
\[
yx - y = x - 2
\]
\[
yx - x = -y - 2
\]
\[
x(y - 1) = -y - 2
\]
\[
x = \frac{-y - 2}{y - 1}
\]
Thus, the inverse of \( g(x) \) is \( g^{-1}(x) = \frac{-x - 2}{x - 1} \), which is not equal to \( g(x) \). Therefore, \( g(x) \) is not self-inverse.

### 3. \( h(x) = \frac{x + 3}{x - 2} \)
Find the inverse by solving \( y = \frac{x + 3}{x - 2} \) for \( x \):

\[
y(x - 2) = x + 3
\]
\[
yx - 2y = x + 3
\]
\[
yx - x = 2y + 3
\]
\[
x(y - 1) = 2y + 3
\]
\[
x = \frac{2y + 3}{y - 1}
\]
Thus, the inverse of \( h(x) \) is \( h^{-1}(x) = \frac{2x + 3}{x - 1} \), which is not equal to \( h(x) \). Therefore, \( h(x) \) is not self-inverse.

### 4. \( k(x) = \frac{x + 1}{x - 1} \)
Find the inverse by solving \( y = \frac{x + 1}{x - 1} \) for \( x \):

\[
y(x - 1) = x + 1
\]
\[
yx - y = x + 1
\]
\[
yx - x = y + 1
\]
\[
x(y - 1) = y + 1
\]
\[
x = \frac{y + 1}{y - 1}
\]
Thus, the inverse of \( k(x) \) is \( k^{-1}(x) = \frac{x + 1}{x - 1} \), which is exactly \( k(x) \). Therefore, \( k(x) \) **is** self-inverse.

### Conclusion:
The function \( k(x) = \frac{x + 1}{x - 1} \) is the only function that is the same as its inverse.

Would you like any further details or explanation on the process?

Here are 5 related questions to explore further:
1. What properties must a function have to be self-inverse?
2. How do you determine the inverse of a rational function?
3. Can a function have more than one inverse?
4. Are all linear functions self-inverse?
5. How does symmetry relate to self-inverse functions?

**Tip**: If a function has symmetry over the line \( y = x \), it is more likely to be self-inverse.

Which functions are the same as their inverse functions? The functions given are f(x) = (x-1)/(x+5), g(x) = (x-2)/(x-1), h(x) = (x+3)/(x-2), and k(x) = (x+1)/(x-1).

Identify which rational functions are the same as their inverse functions. Explore the inverse property for rational functions with a detailed analysis. Suitable for students in Grades 10-12.

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