# 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:

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

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

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
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Sherica
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Sherica
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Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
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Asali
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Samantha
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Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
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Cesar
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
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Prasenjit
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Azam
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Prasenjit
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Damian
silver nanoparticles could handle the job?
Damian
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Azam
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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