# Introduction and key concepts  (Page 2/4)

 Page 2 / 4

## Relations and functions

In earlier grades, you saw that variables can be related to each other. For example, Alan is two years older than Nathan. Therefore the relationship between the ages of Alan and Nathan can be written as $A=N+2$ , where $A$ is Alan's age and $N$ is Nathan's age.

In general, a relation is an equation which relates two variables. For example, $y=5x$ and ${y}^{2}+{x}^{2}=5$ are relations. In both examples $x$ and $y$ are variables and 5 is a constant, but for a given value of $x$ the value of $y$ will be very different in each relation.

Besides writing relations as equations, they can also be represented as words, tables and graphs. Instead of writing $y=5x$ , we could also say “ $y$ is always five times as big as $x$ ”. We could also give the following table:

 $x$ $y=5x$ 2 10 6 30 8 40 13 65 15 75

## Investigation : relations and functions

Complete the following table for the given functions:

 $x$ $y=x$ $y=2x$ $y=x+2$ 1 2 3 50 100

## The cartesian plane

When working with real valued functions, our major tool is drawing graphs. In the first place, if we have two real variables, $x$ and $y$ , then we can assign values to them simultaneously. That is, we can say “let $x$ be 5 and $y$ be 3”. Just as we write “let $x=5$ ” for “let $x$ be 5”, we have the shorthand notation “let $\left(x,y\right)=\left(5,3\right)$ ” for “let $x$ be 5 and $y$ be 3”. We usually think of the real numbers as an infinitely long line, and picking a number as putting a dot on that line. If we want to pick two numbers at the same time, we can do something similar, but now we must use two dimensions. What we do is use two lines, one for $x$ and one for $y$ , and rotate the one for $y$ , as in [link] . We call this the Cartesian plane . The Cartesian plane is made up of an x - axis (horizontal) and a y - axis (vertical).

## Drawing graphs

In order to draw the graph of a function, we need to calculate a few points. Then we plot the points on the Cartesian Plane and join the points with a smooth line.

Assume that we were investigating the properties of the function $f\left(x\right)=2x$ . We could then consider all the points $\left(x;y\right)$ such that $y=f\left(x\right)$ , i.e. $y=2x$ . For example, $\left(1;2\right),\left(2,5;5\right),$ and $\left(3;6\right)$ would all be such points, whereas $\left(3;5\right)$ would not since $5\ne 2×3$ . If we put a dot at each of those points, and then at every similar one for all possible values of $x$ , we would obtain the graph shown in [link]

The form of this graph is very pleasing – it is a simple straight line through the middle of the plane. The technique of “plotting”, which we have followed here, is the key element in understanding functions.

## Investigation : drawing graphs and the cartesian plane

Plot the following points and draw a smooth line through them.(-6; -8),(-2; 0), (2; 8), (6; 16)

## Notation used for functions

Thus far you would have seen that we can use $y=2x$ to represent a function. This notation however gets confusing when you are working with more than one function. A more general form of writing a function is to write the function as $f\left(x\right)$ , where $f$ is the function name and $x$ is the independent variable. For example, $f\left(x\right)=2x$ and $g\left(t\right)=2t+1$ are two functions.

Both notations will be used in this book.

If $f\left(n\right)={n}^{2}-6n+9$ , find $f\left(k-1\right)$ in terms of $k$ .

1. $\begin{array}{ccc}\hfill f\left(n\right)& =& {n}^{2}-6n+9\hfill \\ \hfill f\left(k-1\right)& =& {\left(k-1\right)}^{2}-6\left(k-1\right)+9\hfill \end{array}$
2. $\begin{array}{ccc}& =& {k}^{2}-2k+1-6k+6+9\hfill \\ & =& {k}^{2}-8k+16\hfill \end{array}$

We have now simplified the function in terms of $k$ .

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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
How can I make nanorobot?
Lily
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
how can I make nanorobot?
Lily
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
Got questions? Join the online conversation and get instant answers! By Subramanian Divya         By