I am Adding this in BIT forum Too.
Did you get an answer. I don't think so.
Just because, you need to do the exercises yourself. Otherwise you will never learn.
By getting someone to do it for you will give you a definite 'F' for the subject
However to give you a start I'll answer the questions (BTW: my memory is not good on these).
You can always ask questions like what is bijection. What is g ? f means.. things like that. But not exercises.
First read
Function composition. Very nice article from wiki.
g ? f is read as g circle f and it does the operation g( f(x) ).
7. I don't see the question here.
However I see something very easy here.
f5 ? f4 ? f3 ? f2 ? f1 = +
since for all x in A,
f5(f4(f3(f2(f1(x))))) = +
f1 and f4 are not onto functions (Also known as non-surjective function). In f1, there is 3 in B and in f4, there is x, y.
f2 and f5 are both one-to-one and onto functions (Also known as
bijective function).
f3 is an onto function (Also known as
Surjective function) but NOT one-to-one.
8.
A is correct
B is correct
C is not correct. Reason: This is not always true since Function composition is not not always commutative.
FYI:
The functions g and f are said to commute with each other if g ? f = f ? g. In general, composition of functions will not be commutative. Commutativity is a special property, attained only by particular functions, and often in special circumstances. But a function always commutes with its
inverse to produce the
identity mapping.
In simple terms, if there is a function f defined as f(x) = x. Then f
-1(x)=x. Such a function is called an identity function. In that case f
-1 ? f = f ? f
-1. The identity function f on M is often denoted by id
M.
However, the composition of functions is always associative. That is, if f, g, and h are three functions with suitably chosen domains and codomains, then f ? (g ? h) = (f ? g) ? h.
11.
(b) and (d) are correct.
f(x) = x
2 + 1
g(x) = 3x - 2
f(g(x)) = f ? g = f(3x - 2)
= (3x - 2)
2 + 1
= (9x
2 - 12x + 4) + 1
= 9x
2 - 12x + 5
g(f(x)) = g ? f = g(x
2 + 1)
= 3(x
2 + 1) - 2
= 3x
2 + 3 - 2
= 3x
2 + 1
I'll answer (12) later... I need to refresh my mind with relations on pairs.