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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 = 5 x 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 = 5 x , we could also say “ y is always five times as big as x ”. We could also give the following table:

x y = 5 x
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 = 2 x 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 ( x , y ) = ( 5 , 3 ) ” 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 ( x ) = 2 x . We could then consider all the points ( x ; y ) such that y = f ( x ) , i.e. y = 2 x . For example, ( 1 ; 2 ) , ( 2 , 5 ; 5 ) , and ( 3 ; 6 ) would all be such points, whereas ( 3 ; 5 ) would not since 5 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]

Graph of f ( x ) = 2 x

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 = 2 x 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 ( x ) , where f is the function name and x is the independent variable. For example, f ( x ) = 2 x and g ( t ) = 2 t + 1 are two functions.

Both notations will be used in this book.

If f ( n ) = n 2 - 6 n + 9 , find f ( k - 1 ) in terms of k .

  1. f ( n ) = n 2 - 6 n + 9 f ( k - 1 ) = ( k - 1 ) 2 - 6 ( k - 1 ) + 9
  2. = k 2 - 2 k + 1 - 6 k + 6 + 9 = k 2 - 8 k + 16

    We have now simplified the function in terms of k .

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
absolutely yes
Daniel
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Akash Reply
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
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SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
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Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
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Source:  OpenStax, Siyavula textbooks: grade 10 maths [ncs]. OpenStax CNX. Aug 05, 2011 Download for free at http://cnx.org/content/col11239/1.2
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