# Introduction and key concepts  (Page 2/4)

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## 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 $k$ .

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
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
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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