Introduction and key concepts

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.