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

Relations

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.

• 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:

There are four 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 1 – more 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)

Functions

Example 2 : f (x) –> x²  OR  f(x) = x² 

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 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.

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