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Functions, Image and Pre-image - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Functions, Image and Pre-image is considered one of the most asked concept.

  • 24 Questions around this concept.

Solve by difficulty

A real-valued function f\left ( x \right ) satisfies the functional equation  f\left ( x -y\right )=f(x)f(y)-f(a-x)f(a+y) where 'a' is a given constant and f(0)=1, \ f(2a-x)  is equal to

Concepts Covered - 1

Functions, Image and Pre-image

Function-

A relation from a set A to a set B is said to be a function from A to B if every element of set A has one and only one image in set B.

OR

A and B are two non-empty sets, then a relation from A to B is said to be a function if each element x in A is assigned a unique element f(x) in B, and it is written as 

f : A ➝ B and read as f is mapping from A to B.

                      

                Function                                        Function                                             Not a function  

 

          Not a function

Third one is not a function because, d is not related(mapped) to any element in B.

Fourth is not a function as element a in A is mapped to more than one element in B. 

 

If f is a function from A to B and (a, b) belongs to f, then f (a) = b, where 'b' is called the image of 'a' under f and 'a' is called the pre-image of 'b' under f.

In the ordered pair (1,2). 1 is the pre-image of 2.


Number of function from A to B

\\\mathrm{Let\;set\;A=\left \{ x_1,x_2,x_3.............,x_ m\right \}\;\;i.e.\;\;m\;\;elements}\\\mathrm{and\;\;B=\left \{ y_1,y_2,y_3.................y_n \right \}\;\;n\;elements}\\

Total number of function from A to B = nm 

(The proof of this formula requires the use of Permutation and Combination, so it will be covered later)
 

Vertical Line Test

Functionality check using the graph:

If any line drawn parallel to y-axis cuts the curve at most one point, then it is a function.

If any such line cuts the graph at more than one point, then it is not a function. 

             


In figure 1, any line parallel to y-axis cuts the curve at one point only. Each value of x would have one and only one image (value of y), so figure 1 is a function.

Whereas in figure 2, a line parallel to y-axis cuts the curve in three points. Here for x = x1 , we have three images i.e. y1, y2, and y3 . Therefore, figure 2 is not a function.

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Functions, Image and Pre-image

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