Problem 100 - Algebra and Trigonometry

Question

Find the inverse of \(f(x)=\frac{x-1}{x+1}\)

Step-by-Step Solution

Answer

The inverse of the function \(f(x) = \frac{x - 1}{x + 1}\) is \(f^{-1}(x) = \frac{-x - 1}{x - 1}\)

Step 1: Write The Given Function

Rewrite the given function: \(f(x) = \frac{x - 1}{x + 1}\)

Step 2: Swap \(x\) and \(y\)

To find the inverse, we replace \(f(x)\) with \(y\) (i.e. let \(y = \frac{x - 1}{x + 1}\)), and then swap \(x\) and \(y\). This gives \(x = \frac{y - 1}{y + 1}\)

Step 3: Isolate \(y\)

First multiply both sides by \(y + 1\), then distribute \(x\) on the right side. The equation becomes: \(x(y + 1) = y - 1\). Simplify further to get \(xy + x = y - 1\), and then rearrange terms to isolate \(y\) on one side making \(xy - y = -x - 1\). By factoring out \(y\), we get \(y(x - 1) = -x - 1\). Finally, divide both sides by \(x - 1\), we get \(y = \frac{-x - 1}{x - 1}\).

Step 4: Rewriting the inverse function

Now, replace \(y\) with \(f^{-1}(x)\). This gives \(f^{-1}(x) = \frac{-x - 1}{x - 1}\)