# Introduction and key concepts  (Page 4/4)

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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:

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

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

who was the first nanotechnologist
k
Veysel
technologist's thinker father is Richard Feynman but the literature first user scientist Nario Tagunichi.
Veysel
Norio Taniguchi
puvananathan
Interesting
Andr
I need help
Richard
anyone have book of Abdel Salam Hamdy Makhlouf book in pdf Fundamentals of Nanoparticles: Classifications, Synthesis
what happen with The nano material on The deep space.?
It could change the whole space science.
puvananathan
the characteristics of nano materials can be studied by solving which equation?
sibaram
synthesis of nano materials by chemical reaction taking place in aqueous solvents under high temperature and pressure is call?
sibaram
hydrothermal synthesis
ISHFAQ
how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
STM - Scanning Tunneling Microscope.
puvananathan
how did you get the value of 2000N.What calculations are needed to arrive at it
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