Functions and Inverse Functions

A function is simply a machine: you feed it an input, and it hands back exactly one output. All we're really doing in this lesson is chaining machines together and running them in reverse. The notation has two pieces worth knowing: $f(g(x))$ (composition: run $x$ through $g$ first, then feed the result into $f$) and $f^{-1}$ (the inverse: the machine that undoes $f$, taking each output back to the input it came from). The equations $f^{-1}(f(x)) = x$ just say "do it, then undo it, and you're back where you started."

Everything in calculus is built on functions, so it's worth making sure these basics feel solid. The derivative is a rule for turning one function into another, the chain rule is about compositions, and inverses give us logarithms and the inverse trig functions.

Domain and range. The domain is every input the formula accepts; the range is every output it produces. Watch for division by zero, even roots of negatives, and logs of non-positive numbers.

Composition chains functions: $(f\circ g)(x) = f(g(x))$. Usually $f(g(x)) \ne g(f(x))$, so read from the inside out. Think of putting on socks and then shoes: same two steps, but the order matters.

One-to-one functions. A function has an inverse only if different inputs always give different outputs. It makes sense: if $f(2)$ and $f(-2)$ are both $4$, then the inverse wouldn't know whether to send $4$ back to $2$ or to $-2$. On a graph: every horizontal line crosses it at most once (the horizontal line test). $x^3$ passes; $x^2$ fails, unless you restrict it to $x \ge 0$.

A common mistake. It's tempting to read $f^{-1}(x)$ as $\frac{1}{f(x)}$, because that's what the exponent $-1$ means on a number. It doesn't mean that here: $f^{-1}$ is the function that undoes $f$. For $f(x) = 3x - 2$, the inverse is $\frac{x + 2}{3}$, not $\frac{1}{3x - 2}$.

Finding an inverse.

1. Write $y = f(x)$.
2. Solve for $x$ in terms of $y$.
3. Swap the names: that's $f^{-1}(x)$.
4. Check that $f(f^{-1}(x)) = x$.

The graph of $f^{-1}$ is the graph of $f$ reflected across the line $y = x$, and the domain and range trade places.

Unit conversions (°C ↔ °F), encryption and decryption, and correcting a camera lens's distortion are all function and inverse pairs: one maps forward, the other undoes it exactly.