# Introduction and key concepts

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These are two ways of writing the domain and range of a function, set notation and interval notation . Both notations are used in mathematics, so you should be familiar with each.

## Set notation

A set of certain $x$ values has the following form:

$x:\mathrm{conditions, more conditions}$

We read this notation as “the set of all $x$ values where all the conditions are satisfied”. For example, the set of all positive real numbers can be written as $\left\{x:x\in \mathbb{R},x>0\right\}$ which reads as “the set of all $x$ values where $x$ is a real number and is greater than zero”.

## Interval notation

Here we write an interval in the form ' lower bracket, lower number, comma, upper number, upper bracket '. We can use two types of brackets, square ones $\left[;\right]$ or round ones $\left(;\right)$ . A square bracket means including the number at the end of the interval whereas a round bracket means excluding the number at the end of the interval. It is important to note that this notation can only be used for all real numbers in an interval. It cannot be used to describe integers in an interval or rational numbers in an interval.

So if $x$ is a real number greater than 2 and less than or equal to 8, then $x$ is any number in the interval

$\left(2;8\right]$

It is obvious that 2 is the lower number and 8 the upper number. The round bracket means 'excluding 2', since $x$ is greater than 2, and the square bracket means 'including 8' as $x$ is less than or equal to 8.

## Intercepts with the axes

The intercept is the point at which a graph intersects an axis. The $x$ -intercepts are the points at which the graph cuts the $x$ -axis and the $y$ -intercepts are the points at which the graph cuts the $y$ -axis.

In [link] (a), the A is the $y$ -intercept and B, C and F are $x$ -intercepts.

You will usually need to calculate the intercepts. The two most important things to remember is that at the $x$ -intercept, $y=0$ and at the $y$ -intercept, $x=0$ .

For example, calculate the intercepts of $y=3x+5$ . For the $y$ -intercept, $x=0$ . Therefore the $y$ -intercept is ${y}_{int}=3\left(0\right)+5=5$ . For the $x$ -intercept, $y=0$ . Therefore the $x$ -intercept is found from $0=3{x}_{int}+5$ , giving ${x}_{int}=-\frac{5}{3}$ .

## Turning points

Turning points only occur for graphs of functions whose highest power is greater than 1. For example, graphs of the following functions will have turning points.

$\begin{array}{ccc}\hfill f\left(x\right)& =& 2{x}^{2}-2\hfill \\ \hfill g\left(x\right)& =& {x}^{3}-2{x}^{2}+x-2\hfill \\ \hfill h\left(x\right)& =& \frac{2}{3}{x}^{4}-2\hfill \end{array}$

There are two types of turning points: a minimal turning point and a maximal turning point. A minimal turning point is a point on the graph where the graph stops decreasing in value and starts increasing in value and a maximal turning point is a point on the graph where the graph stops increasing in value and starts decreasing. These are shown in [link] .

In [link] (a), E is a maximal turning point and D is a minimal turning point.

## Asymptotes

An asymptote is a straight or curved line, which the graph of a function will approach, but never touch.

In [link] (b), the $y$ -axis and line $h$ are both asymptotes as the graph approaches both these lines, but never touches them.

## Lines of symmetry

Graphs look the same on either side of lines of symmetry. These lines may include the $x$ - and $y$ - axes. For example, in [link] is symmetric about the $y$ -axis. This is described as the axis of symmetry. Not every graph will have a line of symmetry.

## Intervals on which the function increases/decreases

In the discussion of turning points, we saw that the graph of a function can start or stop increasing or decreasing at a turning point. If the graph in [link] (a) is examined, we find that the values of the graph increase and decrease over different intervals. We see that the graph increases (i.e. that the $y$ -values increase) from - $\infty$ to point E, then it decreases (i.e. the $y$ -values decrease) from point E to point D and then it increases from point D to + $\infty$ .

## Discrete or continuous nature of the graph

A graph is said to be continuous if there are no breaks in the graph. For example, the graph in [link] (a) can be described as a continuous graph, while the graph in [link] (b) has a break around the asymptotes which means that it is not continuous. In [link] (b), it is clear that the graph does have a break in it around the asymptote.

## Domain and range

1. The domain of the function $f\left(x\right)=2x+5$ is -3; -3; -3; 0. Determine the range of $f$ .
2. If $g\left(x\right)=-{x}^{2}+5$ and $x$ is between - 3 and 3, determine:
1. the domain of $g\left(x\right)$
2. the range of $g\left(x\right)$
3. On the following graph label:
1. the $x$ -intercept(s)
2. the $y$ -intercept(s)
3. regions where the graph is increasing
4. regions where the graph is decreasing
4. On the following graph label:
1. the $x$ -intercept(s)
2. the $y$ -intercept(s)
3. regions where the graph is increasing
4. regions where the graph is decreasing

#### Questions & Answers

Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
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many many of nanotubes
Porter
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what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
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what is nanomaterials​ and their applications of sensors.
what is nano technology
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