FUNCTIONS (Domain & Range)
A function is defined as a relation in which every pre-image in the pre-image set must have one and only one image in the image set.
Examples of cases when relations are not functions
CASE 1: When one pre-image has multiple images
CASE 2: When all pre-image does not have an image
TYPES OF FUNCTIONS
One-One functions: A function is said to be a one-one function if each pre-image points to a unique image. As illustrated in the following diagram.
Many-one function: When many pre-images points to a single image it is called as a many-one function. The function can be depicted as
It is also known as injective function.
Here C and D both have same image ‘4’
Into function: A function is said to be an into a function if at least one image in the image set has got no pre-image
It can be depicted as
Here 3 doesn’t have a pre-image.
Onto function: A function is said to be an onto function if all the images or elements in the image set has got a pre-image.
Bijective function: A function is said to be a bijective function if it is both a one-one function and an onto function.
NOTE: For the inverse of a function to exist, it must necessarily be a bijective function.
INVERSE OF A FUNCTION
The necessary condition for a function to have an inverse is that it must be a bijective function
NOTE: If f(x)=g(x)+h(x), if g(x) or h(x) or both are one-one function then f(x) is one-one function…(i)
If f(x)=g(x)+h(x), if and only iff both g(x) and h(x) is onto function then f(x) is onto function…. (ii)
5. Check whether inverse of the following function exists?
Ans.
The given function f(x)=x2 + x
ONE-ONE checking
Definitely, on a set of real numbers for every value of x, we will get an f(x) so it supports the definition of being a function
……………………………………………………………………………………………………………………………………..
Now let us consider f(x)=u(x)+v(x)
u(x)=x2
v(x)=x2
Now u(x) is not one-one because using the general idea we can determine the value of x=2 and x=-2.
Hence, we need to check for v(x)=x
Now v(x) is definitely a one-one function
Thus, from postulate (i) we get f(x) is a one-one function.
……………………………………………………………………………………………………………………………………….
ONTO function
Definitely u(x) is onto function. Also v(x) is an onto function. Thus, from postulate (ii) f(x) is an onto function.
Thus, it satisfies the definition of bijective function hence it has got inverse function.
COMPOSITION OF FUNCTIONS
The composition of functions also called as a function of a function is defined as a function which depends on another function to obtain its pre-image set
For illustration Suppose f(g(x))= Then g(x) has an image set. The image set of g(x) becomes the pre-image set of f(g(x)).
Pictorial representation is as shown
The first set of relation diagram is f(g(x)) whereas the last 2 relation set gives g(x) only
NOTE: f(g(x))=fog(x)
6.For f(x) = 2x + 3 and g(x) = -x2 + 1, find the composite function defined by (fog)(x) Soln:
Ans.
(fog)(x) = f(g(x))
= 2 (g(x)) + 3
= 2( -x 2 + 1 ) + 3
= - 2x 2 + 5
7. Given f(2) = 3, g(3) = 2, f(3) = 4 and g(2) = 5, evaluate (fog)(3)
Ans.
(fog)(3) = f(g(3))
(fog)(3) = f(2) = 3
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