Inverse Functions

If a function takes $x$ to $y$, its inverse takes that $y$ back to the original $x$.

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.

When an inverse is a function

For $f^{-1}$ to be a function, every output of $f$ must come from only one input. Such a function is called one-to-one.

The horizontal line test checks this on a graph: if any horizontal line crosses the graph more than once, the original function has repeated outputs, so its inverse relation would assign one input to multiple outputs.

For example, $f(x)=x^2$ on all real numbers is not one-to-one because $f(3)=f(-3)=9$. Restricting the domain to $x\ge0$ makes an inverse function possible.

Domain and range swap

Because inputs and outputs exchange roles,

and

This is essential when an original function has restrictions. The inverse cannot accept values that the original function never produced.

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 $y=x$.

Verifying by composition

True inverses undo one another:

for every $x$ in the inverse's domain, and

for every $x$ in the original domain. Checking both directions can reveal a lost restriction or incorrect branch of a square root.

Inverses in context

Inverse functions reverse conversions and models. If a formula converts Celsius to Fahrenheit, its inverse converts Fahrenheit back to Celsius. If a revenue function gives revenue from quantity, an inverse may estimate quantity from a permitted revenue value.

Units swap too: a function measured in dollars per item may be reversed to items per permitted dollar outcome, depending on the model.

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.

Quick self-check

• Is the original function one-to-one on its stated domain?
• Did I swap $x$ and $y$ and solve for the new output?
• Have domain and range exchanged?
• Do the graphs reflect across $y=x$?
• Do both compositions simplify to $x$ on the proper domains?

Related topics

• Function Composition
• Function Transformations
• Introduction to Functions
• Square Root Function