Functions, Domain & Range

At a glance 11 formulas

1. 1Composition$(f \circ g)(x) = f(g(x))$$g \text{ acts first; } f(g(x)) \ne g(f(x)) \text{ in general}$
2. 2DomainAn even root needs what is under it $\ge 0$, a denominator must be $\ne 0$, and a log's input must be $> 0$. The domain is the overlap of all of them.
3. 3RangeWrite $y = f(x)$ and solve for $x$: the range is every $y$ that gives a real $x$ from the domain.
4. 4Inverse$y = f(x) \iff x = f^{-1}(y)$$f(f^{-1}(x)) = x$
5. 5Line over line$y = \dfrac{ax + b}{cx + d}:\ \text{every } y \ne \dfrac{a}{c}$$d = -a \Rightarrow f(f(x)) = x$
6. 6A line applied twice$f(x) = ax + b \Rightarrow f(f(x)) = a^2x + (a + 1)b$
7. 7Recover f from f(g(x))Put $t = g(x)$, write $x$ in terms of $t$, substitute, then rename $t$ as $x$. Or test one value of $x$ against the options.
8. 8f(x) with f(1/x)Write the equation again with $x$ replaced by $\frac{1}{x}$, and solve the two like simultaneous equations.
9. 9Functional equationsTry $x = 0$, $y = 0$ or $y = 1$ first: it pins down $f(0)$, or a step rule giving $f(x + 1)$ from $f(x)$.
10. 10Max and min$\max(f, g)$ follows the higher graph and $\min(f, g)$ the lower. The max of a rising and a falling line is least where they cross.
11. 11Where graphs meetTwo graphs meet at the real roots of $f(x) - g(x) = 0$; read the heights off the simpler function.