Finding the Inverse of a Function

How to find the inverse of a function

The inverse function $f^{-1}(x)$ "undoes" $f(x)$. If $f$ takes input $a$ and gives output $b$, then $f^{-1}$ takes input $b$ and returns $a$. The process: swap $x$ and $y$, then solve for $y$.

Step-by-step solution: Find the inverse of $f(x) = \dfrac{2x + 3}{x - 1}$

Step 1 — Replace $f(x)$ with $y$:

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

Step 2 — Swap $x$ and $y$ (this is the key step — every input becomes an output and vice versa):

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

Step 3 — Solve for $y$. Multiply both sides by $(y - 1)$:

$x(y - 1) = 2y + 3$

Expand:

$xy - x = 2y + 3$

Collect all $y$-terms on one side:

$xy - 2y = x + 3$

Factor out $y$:

$y(x - 2) = x + 3$

Divide:

$y = \frac{x + 3}{x - 2}$

Step 4 — Write the answer: $f^{-1}(x) = \dfrac{x + 3}{x - 2}$

Step 5 — Verify by checking $f(f^{-1}(x)) = x$:

$f\left(\frac{x+3}{x-2}\right) = \frac{2 \cdot \frac{x+3}{x-2} + 3}{\frac{x+3}{x-2} - 1} = \frac{\frac{2x+6+3x-6}{x-2}}{\frac{x+3-x+2}{x-2}} = \frac{5x}{5} = x \checkmark$

Graphical check

The graph of $f^{-1}$ is the reflection of $f$ across the line $y = x$. Every point $(a, b)$ on $f$ corresponds to $(b, a)$ on $f^{-1}$.

Common mistakes

• Forgetting to swap $x$ and $y$. If you just solve $y = \frac{2x+3}{x-1}$ for $x$, you get $x$ as a function of $y$ — but then you need to relabel $y \to x$ for the inverse.
• Domain restrictions. $f$ has domain $x \neq 1$; $f^{-1}$ has domain $x \neq 2$. The domain of $f^{-1}$ equals the range of $f$.

Try it in the visualization

Both $f(x)$ and $f^{-1}(x)$ are graphed with the $y = x$ mirror line. Point pairs $(a, b) \leftrightarrow (b, a)$ are connected by dashed lines. Drag the test point to verify $f(f^{-1}(x)) = x$.