Question 1: Define a function as a set of ordered pairs.
Answer:
A function is a set of ordered pairs in which no two different ordered pairs have the same coordinate. An equation that produces such a set of ordered pairs defines a function.
Question 2: Define a function as a correspondence between two sets.
Answer:
A function is a special type of relation where every input has a unique output. Definition: A function is a correspondence between two sets (called the domain and the range) such that to each element of the domain, there is assigned exactly one element of the range.
Question 3: What is the fundamental difference between a relation and a function ? Is every relation a function ?
Answer:
Relation Two or more sets can be related to each other by any means. Consider for an example two sets and having and elements respectively, we can have a relation with any ordered pair which shows a relation between the two sets.
Functions A functions can have the same Range mapped as that of in Relation, such that a set of inputs is related with exactly one output.
Consider for an example Set and Set are related in a manner that all the elements of Set are related to exactly one element of Set or many elements of set are related to one element of Set . Thus this type of relation is said to be a function.
It is to be noted that a function cannot have One to Many Relation between the set and .
Relations and Functions Differences:
Differentiating Parameter  Relations  Functions 
Definition  A relation is a relationship between sets of values. Or, it is a subset of the Cartesian product  A function is a relation in which there is only one output for each input. 
Denotation  A relation is denoted by  A function is denoted by or . 
Example 
** It is not function as is input for both and . 

Note:  Every relation is not a function.  Every function is a relation. 
Question 4: Let and and be a function defined by . Find: (i) range of i.e. (ii) preimages of and .
Answer:
Given and
Therefore
i)Range
ii) Let be preimage of
Since , there is no preimage of .
Similarly, let be the preimage of
Since and are preimage of
Similarly, let be the preimage of
Since is preimage of
Therefore preimages of , and is respectively.
Question 5:If a function be defined by
Find: .
Answer:
Given
Question 6: A function is defined by . Determine: (i) range of (ii) (iii)
Answer:
Given … … … … … i)
i) Range (set of all real numbers greater than or equal to )
ii) We have … … … … … ii)
From i) and ii) we get
iii) … … … … … iii)
Question 7: Let , where is the set of all positive real numbers, be such that Determine: (i) the image set of the domain of (ii) (iii) whether holds.
Answer:
Given and … … … … … i)
i) Since , Therefore the image set of the domain(f) = R
ii) Since … … … … … ii)
From i) and ii) we get
iii)
Hence proved.
Question 8: Write the following relations as sets of ordered pairs and find which of them are functions:
(i) .
(ii)
(iii)
Answer:
i) We have
Putting in we get respectively.
Yes, it is a function.
ii) We have
Putting in , we get respectively.
It is not a function from to because two ordered paired in have same first element.
iii) We have
Now , for we get respectively.
Yes, this is a relation.
Question 9: Let and be two functions defined as and . Are they equal functions?
Answer:
We have and
Domain
Domain
Since Domain Domain
and are not equal functions.
Question 10: If are three functions defined from to as follows: (i) (ii) (iii) Find the range of each function.
Answer:
i) We have
Range of set of all real numbers greater than or equal to where
ii) We have
Range
iii) We have
Range where
Question 11: Let and
Determine which of the following sets are functions from to
(i) (ii)
(iii)
Answer:
Given and
i) We have
is a function from to .
ii) We have
is not a function from to because there is an element which is not associated to any element of
iii) We have
is not a function from to because an element is associated to two elements and in
Question 12: Let and be a function given by highest prime factor of . Find range of .
Answer:
Given and be defined as highest prime factor of
highest prime factor of
highest prime factor of
highest prime factor of
highest prime factor of
highest prime factor of
highest prime factor of
Therefore Range
Question 13: If be defined by , then find and .
Answer:
If is such that , then
In other words, is a set of preimages of
Let
Similarly,
Clearly, there is no solution available in
Question 14: Let and . Which of the following relations from to is not a function?
(i) (ii) (iii) (iv)
Answer:
Given and
i)
Therefore is a function [ As it has unique image in for all elements in ]
ii)
is a function [ As it has unique image in for all elements in ]
iii)
is not a function because the elements is associated with two elements and in .
iv)
is a function [ As it has unique image in for all elements in ]
Question 15: Let and let be defined by the highest prime factor of . Find the range of .
Answer:
Given and be defined by the highest prime factor of
highest prime factor of
highest prime factor of
highest prime factor of
highest prime factor of
highest prime factor of
Therefore Range
Question 16: The function is defined by
The relation is defined by
Show that is a function and is not a function.
Answer:
Given
For
and
Therefore we observe that takes unique value at each point in its domain
Therefore is a function.
Also
For
and
Therefore we observe that does not take unique values at each point in its domain
Hence is not a function.
Question 17: If , find
Answer:
Question 18: Express the function given by as set of ordered pairs, where
Answer:
Given given by