Finding the Inverse of a Function II

Problem 257: If $f(x) = \frac{3x -10}{2x + 1}$ , find the inverse function.

Solution: We let $y = f(x)$ .

$f(x) = \frac{3x -10}{2x + 1} \implies y = f(x) = \frac{3x -10}{2x + 1}$

Interchange $x$ and $y$ .

$y = \frac{3x -10}{2x + 1} \implies x & = \frac{3y - 10}{2y + 1}$

Solve for $y$ .

$\begin{align*} x & = \frac{3y - 10}{2y + 1} \\ x(2y + 1) & = 3y - 10 \\ 2xy + x & = 3y - 10 \\ 2xy - 3y & = -10 - x \\ y(2x-3) & = -10 - x \\ y & = \frac{-10-x}{2x-3}\end{align*}$

The new $y$ we call it $f^{-1}(x)$ . That is,

$f^{-1}(x) = \frac{-10-x}{2x-3}.$

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