1 The function rule

A function can be thought of as a rule which operates on an input and produces an output . This is often illustrated pictorially in two ways as shown in Figure 1. The first way is by using a block diagram which consists of a box showing the input, the output and the rule. We often write the rule inside the box. The second way is to use two sets, one to represent the input and one to represent the output with an arrow showing the relationship between them.

More precisely, a rule is a function if it produces only a single output for any given input. The function with the rule ‘treble the input’ is shown in Figure 2.

Note that with an input of 4 the function will produce an output of 12. With a more general input, $x$ say, the output will be $3x$ . It is usual to assign a letter or other symbol to a function in order to label it. The trebling function in Figure 2 has been given the symbol $f$ .

Task!

Write down the output from the function shown in Figure 3 when the input is

1. $4$ ,
2. $-3$ ,
3. $x$
4. $t$ .

Answer

1. When the input is 4 the output is 26
2. When the input is $-3$ the output is $-23$
3. When the input is $x$ the output is $7x-2$
4. When the input is $t$ the output is $7t-2$ .

Several different notations are used by engineers to describe functions. For the trebling function in Figure 2 it is common to write

$f\left(x\right)=3x$

This indicates that with an input $x$ , the function, $f$ , produces an output of $3x$ . The input to the function is placed in the brackets after the ‘ $f$ ’. $f\left(x\right)$ is read as ‘ $f$ is a function of $x$ ’, or simply ‘ $f$ of $x$ ’, meaning that the value of the output from the function depends upon the value of the input $x$ . The value of the output is often called the ‘value of the function’.

Example 1

State in words the rule defined by each of the following functions:

1. $f\left(x\right)=6x$
2. $f\left(t\right)=6t-1$
3. $g\left(x\right)=x^2-7$
4. $h\left(t\right)=t^3+5$
5. $p\left(x\right)=x^3+5$

Solution

1. The rule for $f$ is ‘multiply the input by 6’.
2. Here the input has been labelled $t$ . The rule for $f$ is ‘multiply the input by 6 and subtract 1’.
3. Here the function has been labelled $g$ . The rule for $g$ is ‘square the input and subtract 7’.
4. The rule for $h$ is ‘cube the input and add 5’.
5. The rule for $p$ is ‘cube the input and add 5’.

Note from Example 1, parts (d) and (e), that it is the rule that is important when describing a function and not the letters used. Both $h\left(t\right)$ and $p\left(x\right)$ instruct us to ‘cube the input and add 5’.

Task!

Write down a mathematical function which can be used to describe the following rules:

1. ‘square the input and divide the result by 2’. Use the letter $x$ for input and the letter $f$ to represent the function.
2. ‘divide the input by 3 and add 7’. Call the function $g$ and call the input $t$ .

Answer

Exercise

State the rule of each of the following functions:

1. $f\left(x\right)=5x$ ,
2. $f\left(t\right)=5t$ ,
3. $f\left(x\right)=8x+10$ ,
4. $f\left(t\right)=7t-27$ ,
5. $f\left(t\right)=1-t$ ,
6. $h\left(t\right)=\frac{t}{3}+\frac{2}{3}$ ,
7. $f\left(x\right)=\frac{1}{1+x}$

Answer