Standard Functions and Their Algebraic Forms

The syllabus expects familiarity with a catalogue of standard functions, each identified by its algebraic form, its domain, its range and its graph shape.

Constant function: $f(x)=k$ for a fixed real $k$, whatever the value of $x$. Domain $\mathbb{R}$, range $\{k\}$; its graph is a horizontal straight line at height $k$. It is many-one (unless the domain has one element) and into.

Identity function: $f(x)=x$. Domain and range both $\mathbb{R}$; its graph is the line $y=x$ through the origin at $45^\circ$. It is bijective on $\mathbb{R}$.

Polynomial function: $f(x)=a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0$, where the coefficients are real and $n$ is a non-negative integer. Its domain is always the whole of $\mathbb{R}$. Constant ($n=0$), linear $f(x)=mx+c$ ($n=1$) and quadratic $f(x)=ax^2+bx+c$, $a\neq 0$ ($n=2$) are the simplest cases. A non-zero-slope linear function $\mathbb{R}\to\mathbb{R}$ is bijective; a quadratic over all of $\mathbb{R}$ is always many-one, since the two sides of its parabola repeat every output except at the vertex.

Rational function: a ratio of two polynomials, $f(x)=\dfrac{p(x)}{q(x)}$, defined wherever $q(x)\neq 0$. Its domain always excludes every root of $q(x)$.

Modulus (absolute value) function: $f(x)=|x|$, defined piecewise as $x$ when $x\ge 0$ and $-x$ when $x<0$. Domain $\mathbb{R}$, range $[0,\infty)$ — it is never negative; its graph is a "V" with vertex at the origin. It is an even function.

Signum function: $\operatorname{sgn}(x)$, defined as $-1$ when $x<0$, $0$ when $x=0$, and $+1$ when $x>0$. Domain $\mathbb{R}$, range $\{-1,0,1\}$; its graph is two horizontal rays at heights $-1$ and $+1$ with a single point at the origin. It reports only the sign of its input. …