## 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.

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