# 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] .

## 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.

how to know photocatalytic properties of tio2 nanoparticles...what to do now
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Maciej
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is Bucky paper clear?
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
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Do you know which machine is used to that process?
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how to fabricate graphene ink ?
for screen printed electrodes ?
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What is lattice structure?
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or in general
Ebrahim
in general
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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
what is biological synthesis of nanoparticles
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Cied
types of nano material
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Stotaw
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anybody can imagine what will be happen after 100 years from now in nano tech world
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Uday
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can nanotechnology change the direction of the face of the world
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