# Introduction and key concepts  (Page 3/4)

 Page 3 / 4

If $f\left(x\right)={x}^{2}-4$ , calculate $b$ if $f\left(b\right)=45$ .

1. $\begin{array}{ccc}\hfill f\left(b\right)& =& {b}^{2}-4\hfill \\ \hfill \mathbb{but}f\left(b\right)& =& 45\hfill \end{array}$
2. $\begin{array}{ccc}\hfill {b}^{2}-4& =& 45\hfill \\ \hfill {b}^{2}-49& =& 0\hfill \\ \hfill b& =& +7\phantom{\rule{1.em}{0ex}}\mathbb{or}\phantom{\rule{1.em}{0ex}}-7\hfill \end{array}$

## Recap

1. Guess the function in the form $y=...$ that has the values listed in the table.
 $x$ 1 2 3 40 50 600 700 800 900 1000 $y$ 1 2 3 40 50 600 700 800 900 1000
2. Guess the function in the form $y=...$ that has the values listed in the table.
 $x$ 1 2 3 40 50 600 700 800 900 1000 $y$ 2 4 6 80 100 1200 1400 1600 1800 2000
3. Guess the function in the form $y=...$ that has the values listed in the table.
 $x$ 1 2 3 40 50 600 700 800 900 1000 $y$ 10 20 30 400 500 6000 7000 8000 9000 10000
4. On a Cartesian plane, plot the following points: (1;2), (2;4), (3;6), (4;8), (5;10). Join the points. Do you get a straight line?
5. If $f\left(x\right)=x+{x}^{2}$ , write out:
1. $f\left(t\right)$
2. $f\left(a\right)$
3. $f\left(1\right)$
4. $f\left(3\right)$
6. If $g\left(x\right)=x$ and $f\left(x\right)=2x$ , write out:
1. $f\left(t\right)+g\left(t\right)$
2. $f\left(a\right)-g\left(a\right)$
3. $f\left(1\right)+g\left(2\right)$
4. $f\left(3\right)+g\left(s\right)$
7. A car drives by you on a straight highway. The car is travelling 10 m every second. Complete the table below by filling in how far the car has travelledaway from you after 5, 10 and 20 seconds.
 Time (s) 0 1 2 5 10 20 Distance (m) 0 10 20
Use the values in the table and draw a graph of distance on the $y$ -axis and time on the $x$ -axis.

## Characteristics of functions - all grades

There are many characteristics of graphs that help describe the graph of any function. These properties will be described in this chapter and are:

1. dependent and independent variables
2. domain and range
3. intercepts with axes
4. turning points
5. asymptotes
6. lines of symmetry
7. intervals on which the function increases/decreases
8. continuous nature of the function

Some of these words may be unfamiliar to you, but each will be clearly described. Examples of these properties are shown in [link] .

## Dependent and independent variables

Thus far, all the graphs you have drawn have needed two values, an $x$ -value and a $y$ -value. The $y$ -value is usually determined from some relation based on a given or chosen $x$ -value. These values are given special names in mathematics. The given or chosen $x$ -value is known as the independent variable, because its value can be chosen freely. The calculated $y$ -value is known as the dependent variable, because its value depends on the chosen $x$ -value.

## Domain and range

The domain of a relation is the set of all the $x$ values for which there exists at least one $y$ value according to that relation. The range is the set of all the $y$ values, which can be obtained using at least one $x$ value.

If the relation is of height to people, then the domain is all living people, while the range would be about 0,1 to 3 metres — no living person can have a height of 0m, and while strictly it's not impossible to be taller than 3 metres, no one alive is. An important aspect of this range is that it does not contain all the numbers between 0,1 and 3, but at most six billion of them (as many as there are people).

As another example, suppose $x$ and $y$ are real valued variables, and we have the relation $y={2}^{x}$ . Then for any value of $x$ , there is a value of $y$ , so the domain of this relation is the whole set of real numbers. However, we know that no matter what value of $x$ we choose, ${2}^{x}$ can never be less than or equal to 0. Hence the range of this function is all the real numbers strictly greater than zero.

what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
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
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
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