Introduction and key concepts  (Page 2/4)

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

 $x$ $y=5x$ 2 10 6 30 8 40 13 65 15 75

Investigation : relations and functions

Complete the following table for the given functions:

 $x$ $y=x$ $y=2x$ $y=x+2$ 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 $\left(x,y\right)=\left(5,3\right)$ ” 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 .

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

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

Both notations will be used in this book.

If $f\left(n\right)={n}^{2}-6n+9$ , find $f\left(k-1\right)$ in terms of $k$ .

1. $\begin{array}{ccc}\hfill f\left(n\right)& =& {n}^{2}-6n+9\hfill \\ \hfill f\left(k-1\right)& =& {\left(k-1\right)}^{2}-6\left(k-1\right)+9\hfill \end{array}$
2. $\begin{array}{ccc}& =& {k}^{2}-2k+1-6k+6+9\hfill \\ & =& {k}^{2}-8k+16\hfill \end{array}$

We have now simplified the function in terms of $k$ .

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