Inverse of a Rational Function

Inverting a rational function needs one extra move beyond the usual swap-and-rearrange: after clearing the fraction, $x$ appears in two places, so you must collect the $x$-terms and factor before you can isolate it.

The function $f$ is defined by $f(x) = \dfrac{2x + 1}{x - 3}$ for $x \neq 3$. Find $f^{-1}(x)$. [4]

The working

Step 1, set $y = f(x)$ and clear the fraction: $y = \frac{2x + 1}{x - 3} \;\Rightarrow\; y(x - 3) = 2x + 1 \quad \text{(M1)}$

Step 2, expand and gather $x$-terms on one side: $yx - 3y = 2x + 1 \;\Rightarrow\; yx - 2x = 3y + 1 \quad \text{(M1)}$

Step 3, factor out $x$ and divide: $x(y - 2) = 3y + 1 \;\Rightarrow\; x = \frac{3y + 1}{y - 2} \quad \text{(A1)}$

Step 4, rename ($y \to x$): $f^{-1}(x) = \frac{3x + 1}{x - 2} \quad \text{(A1)}$

Where the marks are won and lost

• Collect all $x$-terms on one side ($yx - 2x$) and everything else on the other ($3y + 1$). This is the step that makes isolating $x$ possible.
• Factor out $x$: $x(y - 2)$. Without factoring, you can’t divide to isolate it.
• Rename at the end, leaving the answer in terms of $y$ is incomplete.

Common mistakes

• Failing to gather the $x$-terms, so $x$ can’t be isolated.
• Sign errors moving $2x$ and $3y$ across.
• Forgetting to rename $y \to x$ for the final inverse.

Full method: Inverse Functions notes. See also Self-Inverse Function. Topic home: Functions pillar.