4  Basic Concepts You Need To Learn About Functions

Basically there are four concepts you need to master in this chapter:

  1. Relations
  2. Functions
  3. Composite Functions
  4. Inverse Functions


Example 1: P is the set of positive even number less than 10.
Answer: P = {2,4,6,8}

It could be explained that the elements for the set of { 2,4,6,8 } shared the common characteristics of positive even number which is less than 10.

We can say that relations exist between set X and Y, when elements of set X was mapped to elements of set Y.

Function - arrow diagram

Arrow Diagram

  • Set X is the domain of the relation, and 1,2,3 is the object
  • Set Y is the codomain of the relation, and 1,4,9 is the image
  • The range is the subset of codomain which contains all the images that have been mapped, in this case the range = {1,4,9}
  • We can also write it as a set of ordered pairs = {(1,1),(2,4),(3,9)}
  • or represent the relation using a graph as below:

Relation in Graph

There are four types of relations:

Types of Relations

Types of Relations

  • 1 to 1 – 1 object being mapped to 1 image
  • 1 to many – at least 1 object having more than 1 image (1 object to many images)
  • many to 1more than 1 object being mapped to same image (many objects to 1 image)
  • many to many – at least 1 object having more than 1 image, and more than 1 object being mapped to the same image. (many objects to many images or vice versa)


Example 2 : f (x) –> xOR  f(x) = x

function machine

Function machine by Wvbailey

We can explain the function by using diagram above. A function f take x=3 as input and output as 9. In short, Function f which is x2 can be treated as a “machine” that converts the input (3)  into the output (9).

Only 1 to 1 and many to 1 relations are functions

Composite Functions

Composite function diagram

Composite Function Diagram by Wvbailey

Composite functions could be explained as two functions f and g being combined, and become g[f(x)]. Refer diagram above, it can be visualized as the combination of two “machines”. The first “machine” takes input x and outputs f(x). The second “machine” takes f(x) and outputs g(f(x)).

In this case, 3 –> [machine 1] –> 9 –> [machine 2] –> 10

From the diagram, we know that f (x) –> x2 and g(x) –> x + 1, so we can find out the composite function of gf as below:

gf(x) = g[f(x)]
= g(x2 )
= x2 + 1

if 3 given as the input (x=3), then

gf(3) = 32 + 1
= 10

Inverse Functions

If given a function f(x) = y, then the inverse function is f -1 (y) = x.

Inverse Function example

How to convert a function into inverse function

We will discuss some of the questions which are commonly come out in the SPM exam in the topic SPM Questions for Functions.


About Cedric Low

MBA holder from University of Southern Queensland Australia (USQ) in 2009. Studied for IT and graduated from University of Hertfordshire in UK back to 2005. A system engineer turned to become a full time Maths tutor and researcher. Living in Malaysia. Editor of

Posted on 02/06/2011, in Functions, SPM Additional Maths and tagged , , , . Bookmark the permalink. Leave a comment.

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