Inverse Functions

Undoing a function

An inverse reverses the complete action of a function. If $f(4)=11$, then

The notation $f^{-1}$ means inverse function. It does not mean the reciprocal $1/f(x)$. An inverse swaps the roles of input and output.

Finding an inverse algebraically

Use a reliable sequence:

1. Write $y=f(x)$.
2. Exchange $x$ and $y$.
3. Solve the new equation for $y$.
4. Rename $y$ as $f^{-1}(x)$.
5. State any necessary domain restriction.

Swapping first makes the reversal visible instead of relying on memorized shortcuts.

Worked example: a linear inverse

The original multiplies by $3$ and subtracts $5$; the inverse adds $5$ and divides by $3$, in reverse order.

Worked example with a restriction

Let $f(x)=(x-2)^2+1$ with domain $x\ge2$. Swap variables:

Then $(y-2)^2=x-1$. Since the original domain uses the right half of the parabola, take the nonnegative square root:

Thus $f^{-1}(x)=2+\sqrt{x-1}$ with domain $x\ge1$. Without the original restriction, the inverse would not be a function.

Graphical meaning

The graphs of a function and its inverse are reflections across the line $y=x$. Each point $(a,b)$ on $f$ becomes $(b,a)$ on $f^{-1}$.

This explains why domain and range swap, why horizontal and vertical features trade roles, and why the two graphs may intersect on the line $y=x$.

Common mistakes

Treating $f^{-1}(x)$ as $1/f(x)$. Inverse and reciprocal are different ideas.

Solving for $x$ without swapping variables. The final formula must take old outputs as new inputs.

Ignoring the horizontal line test. Not every function has an inverse that is a function.

Using both $\pm$ square-root branches. A domain restriction determines the correct branch.

Forgetting domain and range. The formula alone may be incomplete.