Mathematics · Ch 1 — Sets, Relations and Functions

Ways of Representing Functions

1.6.1

Tabular representation. When the domain elements $x_1,x_2,\dots,x_n$ are explicitly listed, a function can be written as a two-row table of arguments $x_1,\dots,x_n$ against values $y_1,\dots,y_n$ .
Graphical representation. When domain and co-domain are subsets of $R$ , plot the domain along the $x$ -axis and co-domain along the $y$ -axis; $x$ is the argument, $f(x)$ is the value. To read $f(x)$ off a graph: draw a vertical line through $x$ , find where it meets the curve at point $P$ , then a horizontal line through $P$ gives $f(x)$ on the $y$ -axis; reading pre-images of a $y$ -value works the same way with horizontal lines.
Analytical representation. $y=f(x)$ given by a formula -- e.g. $x^3+5$ , $\dfrac{\sin x+\cos x}{x^2+1}$ , $\log x+5\sqrt x$ . The natural domain is the set of $x$ for which the formula is actually defined: $y=x^3+3$ and $y=x^4-2$ have domain $(-\infty,\infty)$ ; $y=\dfrac{x-1}{x+1}$ has domain $R-\{-1\}$ ; $y=\sqrt{4-x^2}$ has domain $[-2,2]$ . Piecewise functions. A function can be defined by different formulas on different pieces of the domain, e.g. $f(x)=\begin{cases}0 & -\infty<x\le-2\\ 2x & -2<x\le3\\ x^2 & 3<x\le\infty\end{cases}$ To evaluate at a point, first find which interval it belongs to, then apply that piece's formula: $f(6)=6^2=36$ (since $6\in(3,\infty)$ ), $f(-1)=2(-1)=-2$ , $f(-5)=0$ . …