Frequently Asked Questions in SPM for Functions

We will discuss some pass year SPM exam questions based on the four concepts :

Relations – SPM 2010
Functions – SPM 2007
Composite Functions – SPM 2007
Composite & Inverse Functions – SPM 2010

Relations – SPM 2010 Paper1 Question 1

Diagram 1 shows the relation between set X and set Y in the graph form.

State

(a)    The relation in the form of ordered pairs.
(b)   The type of the relation
(c)    The range of the relation

(3 marks)

Solution:

a)      Analysis

• The question required you to state the relation in the form of ordered pairs, eg: {(a,b),(c,d)},
• Hence we can state it as {(1,p),(2,r),(3,s),(4,p)}.
• Remember that the element of set X come first, follow by the element from set Y, eg: (element from set X, element from set Y).  

b)      Analysis

• We need to know the objects and images
• Objects – 1,2,3,4 (from set X) 
• Images – p,r,s      (from set Y) 
• 1 & 4 could be mapped to p, so we can state the relation as many to one relation.

c)      Analysis

• The range is all the images being mapped, in this case the range = {p,r,s}
• Take note that q is not included as it does not belong to any object

Functions – SPM 2007 Paper1 Question 2

Given that f : x –> | x – 3 |, find the values of x such that f(x) =5. [2marks]

Solution:

Analysis

Given | x – 3 | and f(x) = 5, meant that you have both positive (5) and negative (-5) image value, so,

x – 3 = 5 or x – 3 = -5

        x = 8 or      x = -2  

Thus, x = 8 or x = -2

Composite Functions – SPM 2007 Paper1 Question 3

The following information is about the function h and the composite function h².

h : x –> ax + b, where a and b are constants, and a > 0h2 : x –> 36x – 35

Find the value of a and b [3marks]

Solution:

Given h(x) = ax + b

h²(x)    = hh(x)

= h( ax + b )

= a( ax + b ) + b [ insert ax + b into the function h(x) again ]

= a²x + ab + b

If we compare both composite functions

h²(x) = 36x – 35

h²(x) = a² x + ab + b

you will see that a² = 36,

a = 6 [ Since a > 0, so we do not accept -6 here ]

ab + b = -35

6b + b = -35 

        7b = -35

          b = -5

So a = 6, b = -5

Composite Functions & Inverse Functions – SPM 2010 Paper1 Question 2

Given the functions g:x –> 2x + 1 and h:x –> 3x + 6, find

(a) g^-1(x)

(b) hg^-1(9)

(3 marks)

Solution:

 

Related content:

• Functions