Show that the function \(f\ :\ N\rightarrow N\) given by f (x) = 2x  is one-one but not onto.

3

(a)

Find gof and fog, if \(f:R\rightarrow R\) and \(g:R\rightarrow R\) are given by: \(f(x)=cosx\) and \(g(x)=3x^{ 2 }\). Show that gof \(\neq \) fog.

3

(a)

Let \(f:N\rightarrow R\) be a function defined as \(f(x)=4x^{ 2 }+12x+15\) Show that \(f:N\rightarrow \) Range f is invertible. Find the inverse off.

3

(a)

Find the number of all one-one functions from the set of A = {1, 2, 3} to itself is a permutation of 1, 2, 3.

3

(a)

Show that the relation R in the set A = {1, 2, 3, 4, 5} given by is even R = {(a, b) : |a - b| is even} is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

3

(a)

If \({ x }^{ 16 }{ y }^{ 9 }={ \left( { x }^{ 2 }+y \right) }^{ 17 },\)prove that \(\frac { dy }{ dx } =\frac { 2y }{ x } \)

3

(a)

If \(y={ \left( { sin }^{ -1 }x \right) }^{ 2 }\), then prove that: \(\left( 1-{ x }^{ 2 } \right) \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -x\frac { dy }{ dx } =2\)

3

(a)