Having worked through this Notebook, you should now be familiar with the idea of the mathematical function and its role in describing relationships between variables.
Below is a summary of some points covered in this Notebook. After reading the summary, try the exercises in the next section.
A function is the mathematical way to describe an action carried out on a variable.
It has an input (i.e. the variable) and an output (i.e. the variable after the action has been carried out on it).
Any function needs a name, usually just one letter, and we define the function by writing the equation showing the variable it acts on and action it represents, e.g. f(x)=2x.
Having defined a function in this way, we can then put any value in as the input and calculate the output using the equation.
With more complicated functions, involving several actions, the mathematical description is easier to manipulate than the English version.
Because any value can be put in for the input variable, the equation allows us to cover all cases at once.
We can relate two variables by choosing one variable to be the input and another to be the output. Then the function describes what action must be done to the first variable to get the second.
We call the input variable the independent variable, because we can vary its value as we like. The output variable is called the dependent variable because its value depends on the other variable.
We looked at an example: how temperature changes as we change the depth below the earth's surface. In this case we found the temperature as a function of the depth.
We can rearrange the equation relating two or more variables, so that the variable which had been the independent variable becomes a dependent variable. This is useful if one or more variables are known, and we need to find the other.