Relations and F...

If $$ \phi (x) = 3
f(\frac{x^2}{3} ) + f(3-x^2) \forall x \in (3,4)$$ where  $$f(x) >0 \forall  x (-3,4)$$ then $$\phi (x)$$ is ____________.

Let $$f: R\rightarrow R; f(x)=2x^3-3(p+2)x^2+12px-7$$, $$-5\leq p \leq 5$$, $$p\in I$$. Then the number of values of p for $$f(x)$$ to be invertible is?

If $$f(0)=5$$, then minimum possible number of values of x satisfying $$f(x)=5$$, for $$x\in [0, 170]$$ is?

A real valued function $$f(x)$$ satisfies the function equation $$f(x-y)=f(x)f(y)-f(a-x)f(a+y)$$ where a is a given constant and $$f(0)=1, f(2a-x)$$ is equal to?

Let $$f\left( x \right)={2^{10}}x + 1$$ and $$g\left( x \right) = {3^{10}}x - 1$$ , If $$\left( {fog} \right)\left( x \right) = x$$ , then $$x$$ is equal to

$$f : R^+ \rightarrow R$$ defined by $$f(x) = 2^x , \, x \in (0, 1), \, f(x) = 3^x , \, x \in [1, \, \infty)$$ is 

Let f(x) and g(x) be the differentiable functions for $$1\le x\le 3$$ such that f(1)=2=g(1) and f(3)=10. Let there exist exactly one real number $$cE (1,3)$$ such that 3f'(c)=g'(c), then the value of g(3) must be

If $$f(x)=|x|$$ and $$g(x)=[x]$$, then value of $$fog \left(-\dfrac {1}{4}\right)+gof \left(-\dfrac {1}{4}\right)$$ is  ?

The function $$f : R\rightarrow R$$ given by, then $$f(x)=3-2\sin x$$ is

If $$f\left( x \right)  = \sin ^{ 2 }{ x } + \sin ^{ 2 }({ { x }+\frac { \pi  }{ 3 })  + \cos { x\cos { \left( { x } + \frac { \pi  }{ 3 }  \right),  ~g(\frac { 5 }{ 4 })  = 1, \text{then} \left( gof \right)\left( x \right)  }  }\ \text{ is}\  \text{equal}\  \text{to}  } $$