Topic 2.8 Notes – Inverse Functions

What an Inverse Function Is

An inverse function undoes another function. If $f(a)=b$, then the inverse sends $b$ back to $a$, so $f^{-1}(b)=a$.

That same idea shows up in every representation:

• Ordered pairs swap. If $(a,b)$ is on $f$, then $(b,a)$ is on $f^{-1}$.
• Domain and range switch. The outputs of $f$ become the inputs of $f^{-1}$, so $\text{domain}(f^{-1})=\text{range}(f)$ and $\text{range}(f^{-1})=\text{domain}(f)$.
• Known values reverse. If $f(5)=-1$, then $f^{-1}(-1)=5$.
• Tables reverse. Swap each input-output pair to build the inverse table.
• Context reverses too. If $f$ takes time and gives amount, then $f^{-1}$ takes amount and gives time. The units switch with the roles.

One common mistake gets tested a lot. Inverse notation is not reciprocal notation:

$$f^{-1}(x)\ne \frac{1}{f(x)}$$

When a Function Has an Inverse

A function is invertible on a given domain when each output comes from exactly one input. That is the same as being one-to-one.

If a function repeats an output, the reverse mapping breaks. One input in the inverse would have to point to two outputs, and that is not a function.

Horizontal line test

A graph is one-to-one if every horizontal line hits it at most once.

This is exactly why a full parabola fails the test, but a restricted branch can pass.

Restricted quadratic and its inverse

Quadratics are the classic example. A parabola over all real numbers is not one-to-one, but you can restrict the domain to one side of the vertex.

• For $y=(x-2)^2+1$, use either $x\ge2$ or $x\le2$.
• Those give different inverses, so the chosen restriction is part of the answer.
• In context, the domain might already be limited, and that can decide which inverse makes sense.

Finding the Inverse

The algebra method is always the same:

1. Write $y=f(x)$
2. Switch $x$ and $y$
3. Solve for $y$
4. Rename it $f^{-1}(x)$
5. State the inverse domain from the original range

Example with a restricted quadratic:

$$f(x)=(x-2)^2+1,\quad x\ge2$$

Swap and solve:

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

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

$$x-1=(y-2)^2$$

$$y-2=\sqrt{x-1}$$

$$f^{-1}(x)=2+\sqrt{x-1}$$

The $+\sqrt{\phantom{x}}$ branch matches $x\ge2$. If the original restriction had been $x\le2$, the inverse would be $2-\sqrt{x-1}$.

The reverse-operations view helps too. Subtract 2, square, add 1 becomes subtract 1, square root, add 2.

How Inverses Look in Different Representations

• Numerical
  Reverse every pair in a table. If original outputs repeat, no inverse function exists.
• Graphical
  The inverse is the reflection across $y=x$. $(a,b)\rightarrow(b,a)$, intercepts trade roles when relevant, and points on $y=x$ stay fixed.
• Analytical
  Swap variables and solve for $y$.
• Verbal and context
  Say clearly what the inverse input means and what the inverse output means, with units.

Illustrative examples you should recognize:

• table of values where pairs are reversed
• graph reflection over $y=x$
• contextual models such as time and amount, or amount and time

Checking and Using the Inverse

A function and its inverse undo each other:

$$f(f^{-1}(x))=x \qquad \text{and} \qquad f^{-1}(f(x))=x$$

Use composition to verify inverses. Domain still matters. With restricted functions, the simplification may only work after using the restriction.

You also use inverses to solve equations. Finding $a$ such that $f(a)=b$ is the same as computing $f^{-1}(b)$.

Common mistakes:

• treating $f^{-1}(x)$ like $\frac1{f(x)}$
• forgetting to restrict a non-one-to-one function
• swapping $x$ and $y$ but not solving for $y$
• keeping both square-root branches
• using the wrong inverse domain
• ignoring context limits