Functions practice · Mixed difficulty

If $f\left( x \right) = \frac{{1 - x}}{{1 + x}}\left( {x \ne - 1} \right)$, then ${f^{ - 1}}\left( x \right)$ equals to

The period of $f(x)=2 \cos \left(\frac{x-\pi}{3}\right)$ is

Let $F:R \to R$ and $g:R \to R$ be two functions defined by $f(x) = {\log _e}({x^2} + 1) - {e^{ - x}} + 1$ and $g(x) = \frac{{1 - 2{e^{2x}}}}{{{e^x}}}.$
Then for which of the following range of $\alpha ,$ the inequality $f\left( {g\left( {\frac{{{{(\alpha - 1)}^2}}}{3}} \right)} \right) > f\left( {g\left( {\alpha - \frac{5}{3}} \right)} \right)$ holds?

Suppose that $f$ is a periodic function with period $\frac{1}{2}$ and that $f(2)=5$ and $f\left(\frac{9}{4}\right)=2$ then $f(-3)-f\left(\frac{1}{4}\right)$ has the value equal to

If $f(x)=\cos (\log x)$, then $f\left(x^{2}\right) f\left(y^{2}\right)-\frac{1}{2}\left[f\left(\frac{x^{2}}{2}\right)+f\left(\frac{x^{2}}{y^{2}}\right)\right]$ has the value

The domain of definition of the function

$f\left( x \right) = {\log _2}\left[ { - {{\left( {{{\log }_2}x} \right)}^2} + 5{{\log }_2}x - 6} \right],$ is

The largest possible set of real numbers which can be the domain of $f\left( x \right) = \sqrt {1 - \frac{1}{x}}$ is

If $f(x)=\cos (\log x)$, then the value of $f(x) \cdot f(4)-\frac{1}{2}\left[f\left(\frac{x}{4}\right)+f(4 x)\right]$

If $f(x)=x^2-1$ and $g(x)=3 x+1$, then $(g \circ f)(x)=$

The maximum value of $4\,\,{\sin ^2}x - 12\,\,\sin \,\,x + 7$ is

The domain of the function $f(x)=\frac{1}{\sqrt{[x]^2-[x]-6}}$ is (where [.] denotes G.I.F)

The domain of definition of
$f\left( x \right) = {\log _{10}}\left\{ {1 - {{\log }_{10}}\left( {{x^5} - 5x + 16} \right)} \right\},$ is

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

The period of $f(x)=x-[x]$, if it is periodic, is

Which pairs of function is identical ?

Let $\mathrm{E}=\{1,2,3,4\}$ and $\mathrm{F}=\{1,2\}$. Then the number of onto functions from $\mathrm{E}$ to $\mathrm{Fis}$

If $f$ and $g$ are two functions defined as $f(x)=x+2$, $x \leq 0 ; g(x)=3, x \geq 0$, then the domain of $f+g$ is

The range of the function $f\left( x \right) = {x^2} - 6x + 7$ is

In a function $f(x)$ is defined for $x \in \left[ {0,\;1} \right],$ then the function $f\left( {2\;x + 3} \right)$ is defined for

Given that $x, y$ and $b$ are real numbers and $x < y, b < 0$, then

The domain of definition of the function

$f\left( x \right) = \sqrt {{{\log }_{10}}\left( {\frac{{5x - {x^2}}}{4}} \right)} ,\mathrm{is}$

The domain of ${\sin ^{ - 1}}\left[ {{{\log }_3}\left( {\frac{x}{3}} \right)} \right]$ is

If $f$ is an exponential function and $g$ is a logarithmic function, then $f \circ g(1)$ will be

Let $\mathrm{A}=\left\{x_{1}, x_{2}, \ldots \ldots \ldots, x_{7}\right\}$ and $\mathrm{B}=\left\{y_{1}, y_{2}, y_{3}\right\}$ be two sets containing seven and three distinct elements respectively. Then the total number of functions $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ that are onto, if there exist exactly three elements $x$ in A such that $\mathrm{f}(x)=y_{2}$, is equal to :

The domain of ${\sin ^{ - 1}}\left[ {{{\log }_2}\left( {\frac{x}{{12}}} \right)} \right]$ is

The domain of the function

$f\left( x \right) = {\log _{10}}\left( {\sqrt {x - 4} + \sqrt {6 - x} } \right)$ is

Let $f: R \rightarrow R$ be defined by $f(x)=3 x-4$, then $f^{-1}(x)$ is:

The domain of the function $f\left( x \right) = {\log _{3 + x}}\left( {{x^2} - 1} \right)$ is

The range of the function $f(x)=|x|$ is

If the domain of the function $f(x)=\frac{[x]}{1+x^{2}}$, where $[x]$ is greatest integer $\leq x$, is $[2,6)$, then its range is