In Review of past work, we were introduced to variables and constants. To recap, a variable can take any value in some set of numbers, so long as the equation is consistent. Most often, a variable will be written as a letter.

A constant has a fixed value. The number 1 is a constant. Sometimes letters are used to represent constants, as they are easier to work with.

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+2A=N+2, where AA is Alan's age and NN is Nathan's age.

In general, a relation is an equation which relates two variables. For example, y=5xy=5x and y2+x2=5y2+x2=5 are relations. In both examples xx and yy are variables and 5 is a constant, but for a given value of xx the value of yy 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=5xy=5x, we could also say “yy is always five times as big as xx”. We could also give the following table:

When working with real valued functions, our major tool is drawing graphs. In the first place, if we have two real variables, xx and yy, then we can assign values to them simultaneously. That is, we can say “let xx be 5 and yy be 3”. Just as we write “let x=5x=5” for “let xx be 5”, we have the shorthand notation “let (x,y)=(5,3)(x,y)=(5,3)” for “let xx be 5 and yy 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 xx and one for yy, and rotate the one for yy, as in Figure 1. We call this the Cartesian plane.

Figure 1: The Cartesian plane is made up of an x-x-axis (horizontal) and a y-y-axis (vertical).

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)=2xf(x)=2x. We could then consider all the points (x;y)(x;y) such that y=f(x)y=f(x), i.e. y=2xy=2x. For example, (1;2),(2,5;5),(1;2),(2,5;5), and (3;6)(3;6) would all be such points, whereas (3;5)(3;5) would not since 5≠2×35≠2×3. If we put a dot at each of those points, and then at every similar one for all possible values of xx, we would obtain the graph shown in Figure 2

Figure 2: Graph of f(x)=2xf(x)=2x

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.

Thus far you would have seen that we can use y=2xy=2x 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)f(x), where ff

Both notations will be used in this book.

Thus far, all the graphs you have drawn have needed two values, an xx-value and a yy-value. The yy-value is usually determined from some relation based on a given or chosen xx-value. These values are given special names in mathematics. The given or chosen xx-value is known as the independent variable, because its value can be chosen freely. The calculated yy-value is known as the dependent variable, because its value depends on the chosen xx-value.

The domain of a relation is the set of all the xx values for which there exists at least one yy value according to that relation. The range is the set of all the yy values, which can be obtained using at least one xx 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 xx and yy are real valued variables, and we have the relation y=2xy=2x. Then for any value of xx, there is a value of yy, so the domain of this relation is the whole set of real numbers. However, we know that no matter what value of xx we choose, 2x2x can never be less than or equal to 0. Hence the range of this function is all the real numbers strictly greater than zero.

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.

The intercept is the point at which a graph intersects an axis. The xx-intercepts are the points at which the graph cuts the xx-axis and the yy-intercepts are the points at which the graph cuts the yy-axis.

In Figure 3(a), the A is the yy-intercept and B, C and F are xx-intercepts.

You will usually need to calculate the intercepts. The two most important things to remember is that at the xx-intercept, y=0y=0 and at the yy-intercept, x=0x=0.

For example, calculate the intercepts of y=3x+5y=3x+5. For the yy-intercept, x=0x=0. Therefore the yy-intercept is yint=3(0)+5=5yint=3(0)+5=5. For the xx-intercept, y=0y=0. Therefore the xx-intercept is found from 0=3xint+50=3xint+5, giving xint=-53xint=-53.

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.

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 Figure 4.

Figure 4: (a) Maximal turning point. (b) Minimal turning point.

In Figure 3(a), E is a maximal turning point and D is a minimal turning point.

An asymptote is a straight or curved line, which the graph of a function will approach, but never touch.

In Figure 3(b), the yy-axis and line hh are both asymptotes as the graph approaches both these lines, but never touches them.

Graphs look the same on either side of lines of symmetry. These lines may include the xx- and yy- axes. For example, in Figure 5 is symmetric about the yy-axis. This is described as the axis of symmetry. Not every graph will have a line of symmetry.

Figure 5: Demonstration of axis of symmetry. The yy-axis is an axis of symmetry, because the graph looks the same on both sides of the yy-axis.

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 Figure 3(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 yy-values increase) from -∞∞ to point E, then it decreases (i.e. the yy-values decrease) from point E to point D and then it increases from point D to +∞∞.

A graph is said to be continuous if there are no breaks in the graph. For example, the graph in Figure 3(a) can be described as a continuous graph, while the graph in Figure 3(b) has a break around the asymptotes which means that it is not continuous. In Figure 3(b), it is clear that the graph does have a break in it around the asymptote.