Answer:
To find the inverse of a function, you typically follow these steps:
1. Replace the function notation with \(y\): \[y = \frac{x/(x-1)}{(x/(x-1))-1}\]
2. Swap the roles of \(x\) and \(y\): \[x = \frac{y/(y-1)}{(y/(y-1))-1}\]
3. Solve for \(y\) in terms of \(x\). This might involve some algebraic manipulation.
Let's work through the steps:
Starting with \[x = \frac{y/(y-1)}{(y/(y-1))-1}\]
First, simplify the expression by getting rid of the fractions. Multiply both sides by \((y-1)\) to clear the fractions:
\[x(y-1) = \frac{y}{y/(y-1)-1}\]
Now, simplify the right side:
\[x(y-1) = \frac{y}{(y-1) - 1}\]
\[x(y-1) = \frac{y}{y - 2}\]
Next, cross-multiply:
\[x(y-1)(y - 2) = y\]
Expand and simplify:
\[x(y^2 - 3y + 2) = y\]
Now, isolate \(y\):
\[xy^2 - 3xy + 2x = y\]
Bring all terms with \(y\) to one side:
\[xy^2 - 3xy - y + 2x = 0\]
Factor out \(y\) on the left side:
\[y(xy^2 - 3x - 1) + 2x = 0\]
Finally, solve for \(y\):
\[y = \frac{2x}{1 - 3x + xy^2}\]
This is the inverse of the original function \(\frac{x/(x-1)}{(x/(x-1))-1}\).
Culled from AI