To find the inverse function of [tex]\( f(x) = \frac{x}{x-2} \)[/tex], let's go through the steps in detail.
1. Start with the given function:
[tex]\[
f(x) = \frac{x}{x-2}
\][/tex]
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = \frac{x}{x-2}
\][/tex]
3. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function:
[tex]\[
x = \frac{y}{y-2}
\][/tex]
4. Cross-multiply to eliminate the fraction:
[tex]\[
x(y - 2) = y
\][/tex]
5. Distribute [tex]\( x \)[/tex] on the left side of the equation:
[tex]\[
xy - 2x = y
\][/tex]
6. Move all terms involving [tex]\( y \)[/tex] to one side:
[tex]\[
xy - y = 2x
\][/tex]
7. Factor out [tex]\( y \)[/tex]:
[tex]\[
y(x - 1) = 2x
\][/tex]
8. Solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{2x}{x - 1}
\][/tex]
So, the inverse function is:
[tex]\[
f^{-1}(x) = \frac{2x}{x-1}
\][/tex]
To match this with the given options:
1. [tex]\( f^{-1}(x)=\frac{2x}{x-1} \)[/tex]
2. [tex]\( f^{-1}(x)=\frac{2x}{x+1} \)[/tex]
3. [tex]\( f^{-1}(x)=-\frac{2x}{x-1} \)[/tex]
4. [tex]\( f^{-1}(x)=\frac{x-2}{x} \)[/tex]
The correct choice is:
[tex]\[
f^{-1}(x)=\frac{2x}{x-1}
\][/tex]
Thus, the answer is the first option.
