To find the inverse of the function [tex]\( f(x) = \frac{x + 1}{x - 2} \)[/tex], we follow a series of steps. Let's denote the inverse function by [tex]\( f^{-1}(x) \)[/tex].
1. Express the function in terms of [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[
y = \frac{x + 1}{x - 2}
\][/tex]
2. Swap [tex]\( y \)[/tex] and [tex]\( x \)[/tex] to solve for [tex]\( y \)[/tex]:
[tex]\[
x = \frac{y + 1}{y - 2}
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
First, multiply both sides by [tex]\( y - 2 \)[/tex] to clear the fraction:
[tex]\[
x(y - 2) = y + 1
\][/tex]
Distribute the [tex]\( x \)[/tex]:
[tex]\[
xy - 2x = y + 1
\][/tex]
Move all terms involving [tex]\( y \)[/tex] to one side and the constant terms to the opposite side:
[tex]\[
xy - y = 2x + 1
\][/tex]
Factor out [tex]\( y \)[/tex] on the left side:
[tex]\[
y(x - 1) = 2x + 1
\][/tex]
4. Isolate [tex]\( y \)[/tex]:
[tex]\[
y = \frac{2x + 1}{x - 1}
\][/tex]
Therefore, the inverse function is:
[tex]\[
f^{-1}(x) = \frac{2x + 1}{x - 1}
\][/tex]
5. Compare with the provided options:
The correct answer is:
[tex]\[
f^{-1}(x) = \frac{2x + 1}{x - 1}
\][/tex]
Hence, the correct choice is:
[tex]\[
\boxed{ f^{-1}(x)=\frac{2x+1}{x-1} }
\][/tex]
This corresponds to the first option provided.
