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

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 a 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 functions from A to B

Let set $A=\left\{x_1, x_2, x_3 \ldots \ldots \ldots \ldots ., x_m\right\}$ i.e. $m$ elements and $B=\left\{\mathrm{y}_1, \mathrm{y}_2, \mathrm{y}_3 \ldots \ldots \ldots \ldots \ldots \mathrm{y}_{\mathrm{n}}\right\} \mathrm{n}$ elements

Total number of functions from A to B = n^m 

(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 the 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 the 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 the y-axis cuts the curve in three points. Here for x = x₁, we have three images i.e. y₁, y_2, and y₃. Therefore, figure 2 is not a function.