Functions · ⁨函数⁩

The vending machine test

• A vending machine takes one code and gives exactly one snack. If code A7 sometimes gives crisps and sometimes chocolate, it's broken.
• A function 函数 works the same way: each input maps to exactly one output. This simple rule underpins all of calculus.

Functions, domain 定义域 and range 值域

• A function $f(x)$ sends each input to exactly one output.
• The domain = allowed inputs; the range = the outputs produced.
• Example: $f(x) = \sqrt{x}$ has domain $x \geq 0$ and range $f(x) \geq 0$.

Composition 复合 and inverses 反函数

• A function is one-one 一一对应 if different inputs always give different outputs — only then does it have an inverse $f^{-1}$.
• Composition $fg(x) = f(g(x))$ — do $g$ first, then $f$.

Finding an inverse

• To find $f^{-1}$: write $y = f(x)$, make $x$ the subject, then swap letters.
• Example: $f(x) = 2x + 6$ → $x = \dfrac{y-6}{2}$ → $f^{-1}(x) = \dfrac{x-6}{2}$.
• The graph of $y = f^{-1}(x)$ is the reflection of $y = f(x)$ in the line $y = x$.

Inverse functions are reflections in $y = x$: the point $(a, b)$ on $f$ becomes $(b, a)$ on $f^{-1}$.

Properties of inverses

• $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ — they undo each other.
• The domain of $f^{-1}$ is the range of $f$, and vice versa.
• Graphically: swap the axes.
• Know the composition of functions $fg(x)$ and graph transformations — a translation, a stretch, and reflections.

Adding to the output or input slides the curve; a multiplier stretches it

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