# Introduction and key concepts  (Page 3/4)

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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] . (a) Example graphs showing the characteristics of a function. (b) Example graph showing asymptotes of a function. The asymptotes are shown as dashed lines.

## 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.

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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