Kinds of Functions by Algebraic Form

Beyond classifying a function by how it matches elements (Section 4), the Business Mathematics & Statistics syllabus also expects familiarity with several standard families of functions, identified by their algebraic form.

Constant function: $f(x) = k$ for a fixed real number $k$, whatever the value of $x$. Its graph is a horizontal straight line at height $k$; it is many-one (unless the domain has only one element) and into (unless the codomain is $\{k\}$ itself).

Linear function: $f(x) = mx + c$, where $m$ (the slope) and $c$ (the intercept) are constants. Its graph is a straight line; if $m \neq 0$, a linear function $f: \mathbb{R} \to \mathbb{R}$ is always bijective.

Quadratic function: $f(x) = ax^2+bx+c$, $a \neq 0$. Its graph is a parabola — opening upward if $a>0$, downward if $a<0$ — and, taken over all of $\mathbb{R}$, is never one-one, because the two sides of the parabola repeat every output value except at the vertex.

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Polynomial function: the general family $f(x)=a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x+a_0$, of which constant, linear and quadratic functions are the simplest special cases ($n=0,1,2$).

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 as $x$ when $x \ge 0$ and $-x$ when $x<0$. Its graph is a 'V' shape with the vertex at the origin, and its range is always $[0,\infty)$ — it is never negative.

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Exponential function: $f(x) = a^x$, for a fixed base $a>0,\ a \neq 1$. Defined for every real $x$, it is always positive, and is strictly increasing when $a>1$ (used, for instance, to model compound growth of an investment or a population). …