Functions Test ...

Let $$f:\{x, y , z\} \rightarrow \{1, 2, 3\}$$ be a one-one mapping such that only one of the following three statements and remaining two are false : $$f(x) \neq 2, f(y) =2, f(z) \neq 1$$, then 

Domain of definition of the function $$f\left( x \right) =\sqrt { 2\sin ^{ -1 }{ \left( 2x \right) +\dfrac { \pi  }{ 3 }  }  }$$, for real value x, is

The domain of definition of the function $$f(x) = \left \{x\right \}^{\left \{x\right \}} + [x]^{[x]}$$ is (where $$\left \{\cdot \right \}$$ represents fractional part and $$[\cdot ]$$ represents greatest integral function).

Let $$f$$ be real valued function defined by $$f(x)=\sin ^{ -1 }{ \left( \cfrac { 1-\left| x \right|  }{ 3 }  \right)  } +\cos ^{ -1 }{ \left( \cfrac { \left| x \right| -3 }{ 5 }  \right)  } $$. Then domain of $$f(x)$$ is given by

$$f:\left( 0,\infty  \right) \rightarrow \left( 0,\infty  \right) $$ 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

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

Domain of the function $$f(x) = \dfrac {x - 3}{(x - 1)\sqrt {x^{2} - 4}}$$ 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

Domain of $$f\left( x \right) =\sqrt { 2{ \left\{ x \right\}  }^{ 2 }-3\left\{ x \right\} +1 }$$ where $$\left\{ . \right\}$$ denotes the fractional part, in $$\left\{ . \right\}$$ is