Introduction and key concepts

Introduction to functions and graphs

Functions are mathematical building blocks for designing machines, predicting natural disasters, curing diseases, understanding world economies and for keeping aeroplanes in the air. Functions can take input from many variables, but always give the same answer, unique to that function. It is the fact that you always get the same answer from a set of inputs that makes functions special.

A major advantage of functions is that they allow us to visualise equations in terms of a graph . A graph is an accurate drawing of a function and is much easier to read than lists of numbers. In this chapter we will learn how to understand and create real valued functions, how to read graphs and how to draw them.

In addition to their use in the problems facing humanity, functions also appear on a day-to-day level, so they are worth learning about. A function is always dependent on one or more variables, like time, distance or a more abstract quantity.

Functions and graphs in the real-world

Some typical examples of functions you may already have met include:-

• how much money you have, as a function of time. You never have more than one amount of money at any time because you can always add everything to give one number. By understanding how your money changes over time, you can plan to spend your money sensibly. Businesses find it very useful to plot the graph of their money over time so that they can see when they are spending too much. Such observations are not always obvious from looking at the numbers alone.
• the temperature is a very complicated function because it has so many inputs, including; the time of day, the season, the amount of clouds in the sky, the strength of the wind, where you are and many more. But the important thing is that there is only one temperature when you measure it in a specific place. By understanding how the temperature is effected by these things, you can plan for the day.
• where you are is a function of time, because you cannot be in two places at once! If you were to plot the graphs of where two people are as a function of time, if the lines cross it means that the two people meet each other at that time. This idea is used in logistics , an area of mathematics that tries to plan where people and items are for businesses.
• your weight is a function of how much you eat and how much exercise you do, but everybody has a different function so that is why people are all different sizes.

Recap

The following should be familiar.

Variables and constants

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.

Investigation : variables and constants

In the following expressions, identify the variables and the constants:

1. $2x^2=1$
2. $3x+4y=7$
3. $y=\frac{-5}{x}$
4. $y=7^x-2$