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} <x \leq 7 \end{cases}

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\}

MA101.2 Functions

The idea of a function is a rule that gives the mapping of an input number to an output number. This function is generally represented by f(x) but nowadays we like the view that the f symbolises the function (intuitively f symbolises the rule the function is applying) and f(x) is actually the number we get when we apply the function or the rule to the number x. (We say f(x) is the value of f at x).

The modern idea of a function only allows rules which assign one value to a particular number of x. For example if f symbolised a rule for which say f(2)=7 and f(2)=4 then f is not a function. Do not confuse this with a rule f which is assigning the same value to two (or more different) numbers. For example f(3)=5 and f(7)=5; this is permitted in our idea of a function.

Also when specifying a function we must not only give the rule, but also give the set of numbers to which the rule is intended to apply. This set of numbers to which the rule applies is called the domain of the function and may be denoted by D(f) or D_f. Thus we say, define the function f by: f(x)=\frac{1}{x},x\in\mathbb{R}\backslash\{^-0\}

Now let f be a function whose domain D(f)=A. Then the set of values that f takes ie \{f(x)|x\in A\} is called the range of f and may be denoted by R(f) or R_f

Some useful notation

  • (a,b) = \{ x \in \mathbb{R} | a < x < b \} – the open interval a,b
  • [a,b] = \{ x \in \mathbb{R} | a \leq x \leq b \} – the closed interval a,b
  • (a,b] = \{ x \in \mathbb{R} | a < x \leq b \}
  • [a,b) = \{ x \in \mathbb{R} | a \leq x< b \}
  • (\infty,a) = \{ x \in \mathbb{R} | x < a \}
  • (\infty,a] = \{ x \in \mathbb{R} | x \leq a \}
  • [-a,\infty) = \{ x \in \mathbb{R} | a \leq x \}
  • (-a,\infty) = \{ x \in \mathbb{R} | a < x \}

Also used is:

  • \mathbb{R}^+ = \{a \in \mathbb{R}, a > 0 \}
  • \mathbb{R}^- =\{a \in \mathbb{R}, a < 0 \}

|x| (read this as mod x where x \in \mathbb{R}) is defined by:

|x| = \begin{cases} x, & \mbox{if } x\geq \mbox{ 0} \\ -x, & \mbox{if } x<\mbox{0} \end{cases}

 


In this course the notation \sqrt{x} where x \geq 0 will always be the positive root, as will also x^\frac{1}{2}.

So, for example \sqrt{4}=2.

Note particularly the effect \sqrt{(-2)^2}=\sqrt{4}=2

In fact generally \sqrt{x^2}=|x|


The so called “floor function” (or integer part function) is notated as, \lfloor x \rfloor and is defined by

\lfloor x \rfloor = \begin{cases} \mbox{integer part of x}\\ \mbox{greatest integer less than or equal to x} \\ \mbox{the greatest integer not bigger than x} \end{cases}

eg.

\lfloor 5.7 \rfloor = 5
\lfloor 3 \rfloor = 3
\lfloor -2.7 \rfloor = -3

MA101.1 Numbers used in mathematics

First a word about numbers used in Mathematics.

\mathbb{N} – the natural numbers.

\mathbb{Z} – the integers.

\mathbb{Q} – the rational numbers (really the ratio numbers). The decimal representation of each rational is either terminating or recurring. Also block recurring decimals are rational. We can represent each rational number as a point on a number line with 0 and 1 marked.

But there are points on the line which are not the representation of a rational number. For example \surd{2} is not rational (proved in the analysis course)  but is on the line.

Now it can be shown (although it is rather hard) that each point of the line corresponds to some decimal that may or may not be block recurring.

\mathbb{R} – the real numbers. The most elementary way of thinking of the reals is as decimals or as all the points on the line. In fact we often speak of the real line. Decimals which are not block recurring are said to be irrational.

For practical purposes the rational numbers are enough because any irrational can be approximated as closely as we like by a rational. For most mathematical purposes however it is convenient to have the irrationals as well.

\mathbb{C} – the complex numbers. These are thought of as numbers of the form a+b\surd{-1} where a and b are real.