Answer:
[tex]f^{-1}(x)=\frac{x}{x-1}[/tex]
Step-by-step explanation:
our function is
[tex]f(x)=\frac{x}{x-1}[/tex]
we are asked to determine the inverse of the above function . In order to do so we put f(x) = y and separate x in terms of y and in the final answer , we replace x with y and x with h(x)
let us see how do we do that
[tex]y=\frac{x}{x-1}[/tex]
multiplying both sides by (x-1)
[tex]y(x-1)=x[/tex]
[tex]xy-y=x[/tex]
subtracting x from both sides and adding y on both sides
[tex]xy-x=y[/tex]
[tex]x(y-1)=y[/tex]
dividing both sides by (y-1)
[tex]x=\frac{y}{y-1}[/tex]
we now replace x with y and x with [tex]f^{-1}(x)
[/tex]
[tex]f^{-1}(x)=\frac{x}{x-1}[/tex]
