MA101.3 Sketching Functions

Graphs of functions may be sketched in the usual way.

Example

Sketch the graph of the function f given by:

$f(x) = \begin{cases} |x| &\mbox{}-2 < x \leq \pi \\ sin(x) -1 &\mbox{} \pi < x < \frac{3 \pi}{2} \\ -2x + 3\pi -2 &\mbox{} \frac{3 \pi}{2}

State the domain and range of f.

$D(f)=(-2,7)\backslash\{\frac{3\pi}{2}\}, R(f)=[0,\pi]\cup[-2,-1]\cup[3\pi-16,-2)$

Any set B which contains the range of a function (ie. $B \supseteq R(f)$) is called a co-domain of the f. The notation $f:A \rightarrow B$ (read: f from A to B) means simply f is a function whose domain is in  A (ie. $A=D(f)$) and whose range is in B (perhaps equal to B).

In practice mathematicians still refer to expressions in x as functions without mentioning any domain. When this is the case it is an unwritten law that one may assume the domain as the largest subset of the reals for which the expression is meaningful.

Example

What are the intended domains when we reference the functions

i) $\frac{1}{x^2-1}$

ii) $cot(x)$

iii) $\frac{x^2-1}{x-1}$

i) $\mathbb{R} \backslash \{-1,1\}$

ii) $\mathbb{R} \backslash \{ k \pi | k \in \mathbb{R} \}$

iii) $\mathbb{R} \backslash \{1\}$