Relations and F...

If  27  $$^{*}$$  3 =243 and  5  $$^{*}$$   4 = 80 .Then what is the value of  3   $$^{*}$$   7 = ?

Consider the function $$f\left( x \right) ={ e }^{ x }$$ and $$g\left( x \right)=\sin ^{ -1 }{ x } $$, then which of the following is/are necessarily true.

Let N be the set of natural numbers and two functions f and g be defined as
and g(n)=$$n-{ \left( -1 \right)  }^{ n }$$ then fog is:

Let $$f:R\rightarrow R$$  defined by $$f\left( x \right) =\frac { { e }^{ { x }^{ 2 } }-{ e }^{ -x^{ 2 } } }{ { e }^{ x^{ 2 } }+{ e }^{ { -x }^{ 2 } } } ,$$ then

Which one of the following is one-one?

The function $$f:R\rightarrow R$$ defined by $$f\left( x \right) =\frac { { e }^{ \left| x \right|  }-{ e }^{ -x } }{ { e }^{ x }+{ e }^{ -x } } $$ is

Let f(x+y)=f f(x) f(y) and  f(x) =1+x g(x) G(x), where $$\underset { x\rightarrow 0 }{ lim } g\left( x \right) =a and \underset { x\rightarrow 0 }{ lim } G\left( x \right) =b,$$ then f' (x) is equal to

Let $$f:R\rightarrow R$$, be defined as $$f(x)={ e }^{ x^{ 2 } }+cosx$$ then f is

$$f:N\rightarrow N\quad where\quad f\left( x \right) =x-{ (-1) }^{ x }$$, then 'f' is

Let (X) be a function satisfying f' (X) = f (X) with f (0) = 1 and g (X) be a function that satisfies f (X) + g (x) = $${ x }^{ 2 },$$ Then the value of the integral $$\int _{ 0 }^{ 1 }{ f } (x)\quad g\quad (x)\quad dx,\quad is$$