Relations and F...

Let $$f: R\rightarrow R$$ be a function such that $$f(x) = ax + 3\sin x + 4\cos x$$. Then $$f(x)$$ is invertible if

$$f:\left( 0,\infty  \right) \rightarrow \left( 0,\infty  \right) $$ is defined by $$f(x)=\begin{cases} { 2 }^{ x },\quad x\in \left( 0,1 \right)  \\ { 5 }^{ x },\quad x\in [1,\infty ) \end{cases}$$ is

In the set $$Q^{+}$$ of all positive rational numbers, the operation $$\ast$$ is defined by the formula $$a\ast b = \dfrac {ab}{6}$$. Then, the inverse of $$9$$ with respect to $$\ast$$ is

Let $$A=\left\{ { a }_{ 1 },{ a }_{ 2 },{ a }_{ 3 },{ a }_{ 4 }{ a }_{ 5 },{ a }_{ 6 } \right\} $$ and $$B=\left\{ { b }_{ 1 },{ b }_{ 2 },{ b }_{ 3 } \right\} $$. The number of functions of $$f:A\rightarrow B$$ such that it is onto and there are exactly three elements in $$A$$ such that $$f(A)=b$$, is

Let $$f(x) = \dfrac {x}{1 - x}$$ and let $$\alpha$$ be a real number. If $$x_{0} = \alpha, x_{1} = f(x_{0}), x_{2} = f(x_{1}), ....$$ and $$x_{2011} = - \dfrac {1}{2012}$$ then the value of $$\alpha$$ is

If $$f:R\rightarrow R$$ is defined by $$f(x)={x}^{2}-6x-14$$, then $$f^{-1}(2)$$ equals to

If $$f:R\rightarrow S$$ defined by
$$f(x)=4\sin { x } -3\cos { x } +1$$ is onto, then $$S$$ is equal to

Let for $$a \neq a_{1} \neq 0,\ f(x)=ax^{2}+bx+c,\ g(x)=a_{1}x^{2}+b_{1}x+c_{1}$$ and $$p(x)=f(x)-g(x)$$. If $$p(x)=0$$ only for $$x=-1$$ and $$p(-2)=2$$, then the value of $$p(2)$$ is

Given that $$f'(x) > g'(x)$$ for all real x, and $$f(0) = g(0)$$. Then $$f(x) < g(x)$$ for all x belongs to