To find the inverse function of [tex]\( f(x) = \frac{x}{x-2} \)[/tex], we need to follow a detailed step-by-step process to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] and then express [tex]\( x \)[/tex] as a function of [tex]\( y \)[/tex].
1. Start with the given function:
[tex]\[
f(x) = \frac{x}{x-2}
\][/tex]
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex] to signify the output of the function:
[tex]\[
y = \frac{x}{x-2}
\][/tex]
3. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. Begin by eliminating the fraction by cross-multiplying:
[tex]\[
y(x-2) = x
\][/tex]
4. Distribute [tex]\( y \)[/tex] on the left-hand side:
[tex]\[
yx - 2y = x
\][/tex]
5. Isolate the [tex]\( x \)[/tex] terms by getting all [tex]\( x \)[/tex] terms on one side of the equation:
[tex]\[
yx - x = 2y
\][/tex]
6. Factor out [tex]\( x \)[/tex] from the left-hand side:
[tex]\[
x(y - 1) = 2y
\][/tex]
7. Solve for [tex]\( x \)[/tex] by dividing both sides by [tex]\( y - 1 \)[/tex]:
[tex]\[
x = \frac{2y}{y - 1}
\][/tex]
8. To find the inverse function [tex]\( f^{-1}(x) \)[/tex], replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] in the expression:
[tex]\[
f^{-1}(x) = \frac{2x}{x - 1}
\][/tex]
Therefore, the inverse function of [tex]\( f(x) = \frac{x}{x-2} \)[/tex] is:
[tex]\[
f^{-1}(x) = \frac{2x}{x - 1}
\][/tex]
