# Formulas, graphs & relations » Working with formulas and functions

## Contents

1. What is a formula?2. How do you work with a formula?

3. What is a function?

4. How do you work with a function?

## 1. What is a formula?

A formula is a mathematical sentence with variables.

You use a formula to describe the relation(ship) between variables.

You use a formula to shortly write a calculation rule.

Calculation rule | Formula |

number of litres times 14 is the number of km you can drive | distance = 14 × number of litres |

number of tickets times 5 minus 50 is the profit | profit = 5 × number of tickets – 50 |

A formula is often written as short as possible. Words in the formula, the variables, are often abbreviated to one letter (preferably no capital letters). Never abbreviate a variable to two or more letters.

In most cases you can also leave out the multiplication sign.

The plural of formula is formulas or formulae.

### Variable

The variables in a formula can be substituted for (changed into) numbers. This way you can calculate with it.

In the formula `amount` = 350 + 80 × `number of months` the variables are `amount` and `number of months`. In the formula `y` = 3`x` + 8 the variables are `y` and `x`.

### Leaving out the multiplication sign

Most often you will work with a multiplication in a formula. The formulas will also increase in length when your maths gets more difficult.

For convenience and for clarity in almost every case you can leave out the multiplication sign. The only exception is, `between two numbers`.

Never remove the multiplication sign between two numbers. 3 × 4 will change into 34. And 3 × 4 is of course 12, not 34.

RULE: The number with which should be multiplied is always in front of the variable.

So when is it allowed?

- A number and a variable: 3 × `a` = 3`a`

- Two variables: `a` × `b` = `ab` *

- A number and a bracket: (3 + `a`) × 5 = 5(3 + `a`)

- A variable and a bracket: `a` × (`a` + 8) = `a`(`a` + 8)

- Two brackets: (`a` + 3) × (`a`^{2} – 4) = (`a` + 3)(`a`^{2} – 4)

- A number and a symbol: 5 × `π` = 5`π`

* This is the reason you may not abbreviate a variable to two letters. `nop` always means `n` × `o` × `p`. For that reason you cannot shorten `Number of pupils` to `nop`.

*Examples*

Formula | Shortened |

3 × a + 8 × a^{2} + 5 |
3a + 8a^{2} + 5 |

a × 7 + 8 × (3 × a + 7) |
7a + 8(3a + 7) |

(7 × b + b^{2} × –4) × –3 |
–3(7b – 4b^{2}) |

Take a look at simplifying formulas to see how you can shorten formulas even further.

## 2. How do you work with a formula?

In your exercise there should always be a formula and a certain value of one the variables given. Fill in the given value into the variable of the formula (change/substitute the variable for the given number). This way you will get a calculation of which you can calculate the answer or an equation you can solve.

*Examples*

Formula | Value | Filled in the value | Answer |

y = 3x + 7 |
x = 4 |
y = 3 × 4 + 7 |
y = 19 |

b = –7 + 6a |
a = –5 |
b = –7 + 6 × –5 |
b = –37 |

m = 7p – 3q |
p = –2 and q = 4 |
m = 7 × –2 – 3 × 4 |
m = –26 |

y = 2x^{2} – 5x |
x = –3^{ } |
y = 2 × (–3)^{2} – 5 × –3Brackets in the power! |
y = 33^{ } |

g = 5h – 8 |
g = 47 |
47 = 5h – 8 |
h = 11See equations |

## 3. What is a function?

A function is nothing more than a more mathematical approach to formulas or calculation rules. This way you can name certain aspects of working with formulas and the formulas itself better.

Most times `f` is a function of `x`. For every possible value of `x` (the *independent variable*) you will have exactly one value of the *dependent variable* `f` (`x`).

The possible values of `x` are called the *arguments* or input of the function; together they form the domain of function `f`.

The associated results of `f` (`x`) are called the *images*; together they form the range of function `f`.

With the function notation you can easily give you formulas different names.

Often `f` (`x`) and `g`(`x`) are used.

But you can also have for example `A`(`t`) and `B`(`t`).

Below you will find a couple of formulas written as a function.

Formula | Function |

b = 3a + 7 |
f (a) = 3a + 7 |

b = –4m + 12 |
g(m) = –4m + 12 |

y = 7x – 3x^{2} |
h(x) = 7x – 3x^{2} |

## 4. How do you work with a function?

This works the same as with formulas, however the notation is slightly different.

*Examples*

Function | Calculate | Filled in value | Answer |

f (x) = 2x + 5 |
f (7) |
f (7) = 2 × 7 + 5 |
f (7) = 19 |

T(h) = 7h^{2} – 4 |
T(3) |
T(3) = 7 × 3^{2} – 4 |
T(3) = 59 |

z(a) = 6a^{2} – 3 |
z(–8)^{ } |
z(–8) = 6 × (–8)^{2} – 3Brackets in the power! |
z(–8) = 381^{ } |

g(t) = –5t + 2 |
g(t) = –8 |
–8 = –5t + 2 |
t = 2See equations |