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Functions, Image and Pre-image is considered one of the most asked concept.
24 Questions around this concept.
A real-valued function satisfies the functional equation
where 'a' is a given constant and
is equal to
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
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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