JEE Functions
This page will teach you how to master JEE Functions. We highlight the main concepts, provide a list of examples with solutions, and include problems for you to try. Once you are confident, you can take the quiz to establish your mastery.
A function from a non-empty set to another non-empty set is a correspondence that assigns to each element of set , one and only one element of set . Mathematically, we write
JEE Conceptual Theory
As per JEE syllabus, the main concepts are definition of function, mappings, even and odd functions, composition of functions and periodic functions.
Definition of function
- Function as a set of ordered pairs
- Domain, co-domain and range
Mappings
- One-one and many-one functions
- Onto and into functions
- Bijective fuctions
Even and odd functions
- Definition and properties
- Even and odd extensions
Composition of functions
Definition and properties
Determining the composite function
Periodic functions
- Properties
- Period and fundamental period
JEE Main Problems
Find the domain of the real-valued function , where represents the greatest integer function.
Concepts tested: domain
Answer:
Solution:
For to be real-valued, has to be positive:
As , excluding the values at which i.e. the domain of the real-valued function isCommon mistakes: It is wrong to consider that because the definition of fractional part function says that .
If , then what is the smallest positive value of such that for all real values of ?
Concepts tested: periodic functions
Answer:
Solution:
As per the definition of the fundamental period, we are asked to find the fundamental period of the function through the statement of the problem. Let . Then since the fundamental period of is the fundamental period of is also .Common mistakes: Thought that was exponential and thus couldn't be periodic? is a counter-example.
Which of following statements is true about the function , where is not an integer multiple of and represents the greatest integer function ?
Concepts tested: even and odd functions
Answer:
Solution:
We have
Since we can say that if is not an integer, it follows that
Hence is an odd function, provided is not an integer multiple of .
Common mistakes: If you thought that , you would get that is neither odd nor even.
JEE Advanced Problems
The base of a tower has the shape of a truncated right circular cone which is surmounted by a right circular cylinder which is again surmounted by a hemisphere of radius . The truncated right circular cone is having a lower base of radius , upper base of radius and height . The cylinder is of radius and height . Suppose that the cross-sectional area of the tower is given by , where is the distance of the cross-section from the lower base of the cone. Then which of the followings is true about ?
Concepts tested: one-one functions
Answer: is one-one on
Solution:
Step 1: Figure out the function
The function is defined in and is given byStep 2: Analyse the function in all the relevant intervals
: decreases on and takes values in .
: is constant on and equals to .
: decreases on and takes values in .
Step 3: Check condition of one-one on all intervals
is decreasing function in the interval and constant function in the interval . Hence, one-one in the interval , but many-one in the interval .Common mistakes: If you didn't figure out the function with respect to the given variable exactly, then you'll come out with inappropriate results.
Let and . Find the range of the function .
Concepts tested: Composition of functions
Answer: D)
Solution:
Step 1: Writing in terms of g(x)
Step 2: Breaking into intervals of
Step 3: Find range for the separate intervals
For : and range of is .
For : and range of is .
For : and range of is .
Step 4: Taking union of all the above intervals of range
Since , hence the range of is all real numbers.
Common mistakes:
- Not having deep understanding of how to figure out the exact composition function .
Once you are confident of Functions, move on to JEE Function Graphs or JEE Limits,Continuity and Differentiability.