Functions practice · Mixed difficulty

$E, I, R, O$ denote respectively the sets of all equilateral, isosceles, right angled and obtuse angled triangles in a plane, then which of the following is not true.

The domain of the function ${\sin ^ - }\left( {{{\log }_2}\frac{{{x^2}}}{2}} \right)$ is

If $f\left( x \right) = {2^x},$ then $f\left( 0 \right),f\left( 1 \right),f\left( 2 \right), \ldots$ are in

If $f(x)$ satisfies the relation $f(x+y)=f(x)+f(y)$ for all $x, y \in R$ and $f(1)=5$ then $f(x)$ is

Let $f(x)=\sin \sqrt{[a]} \times$ (where [] denotes the greatest integer function). If $\mathrm{f}$ is periodic with fundamental period $\pi$, then a belongs to

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

Let $x$ be a non-zero rational number and $y$ be an irrational number. Then $x y$ is

Let $A = \left\{ {x \in R:x \ne 0, - 4 \le x \le 4} \right\}\;and\;f:A \in R$ be defined by $f\left( x \right) = \frac{{\left| x \right|}}{x}\;for\,x \in A.$. Then, the range of $f$ is

Assertion: If $f$ is even function, $g$ is odd function, then $\frac{f}{g}$, $({g} \neq 0)$ is an odd function.

Reason: If $f(-x)=-f(x)$ for every $x$ of its domain, then $f(x)$ is called an odd function and if $f(-x)=f(x)$ for every $x$ of its domain, then ${f}({x})$ is called an even function.

If $f(x)=\frac{x}{x-1}$, then $\frac{f(a)}{f(a+1)}=$

If $f: R \rightarrow R$ satisfies $f(x+y)=f(x)+f(y)$, $\forall x, y \in R$ and $f(1)=7$, then $\sum\limits_{{r=1}}^n f(r)$ is

Let $g(x)=1+x-[x]$ and $f(x)=\operatorname{sgn} x$. Then for all $x, f \circ g(x)$ is equal to

For $x \in\left(0, \frac{3}{2}\right), \operatorname{let} f(x)=\sqrt{x}, g(x)=\tan x$ and $h(x)=\frac{1-x^{2}}{1+x^{2}}$. If $\phi(x)=((h \circ f) \circ g)(x)$, then $\phi\left(\frac{\pi}{3}\right)$ is equal to :

If $f\left( x \right) = \frac{1}{{\sqrt { - x} }}$, then domain of $f\;of$ is

Let $f\left( x \right) = x,g\left( x \right) = 1/x\;and\;h\left( x \right) = f\left( x \right)g\left( x \right).$ Then, $h\left( x \right) = 1,$ if

If $f\left( x \right) = a{\log _e}\left| x \right| + b{x^2} + x$ has extremum at $x = 1\,and\,x = 3$, then

If $f:R \to R\;and\;g:R \to R$ are defined by $f\left( x \right) = x - 3\;and\;g\left( x \right) = {x^2} + 1$ then the values of x for which $g\left\{ {f\left( x \right)} \right\} = 10$ are

Inverse of the function $y=2 x-3$ is

If $f\left( x \right) = {\sin ^2}x$ and the composite function $g\left( {f\left( x \right)} \right) = \left| {\sin x} \right|$, then the function $g\left( x \right)$ is equal to

The domain of definition of the function
$f\left( x \right) = \sqrt {{{\log }_{10}}\left( {\frac{{5x - {x^2}}}{4}} \right)}$ is

The domain for which the functions $f(x)=2 x^2-1$ and $g(x)=1-3 x$ are equal, is

If ${b^2} - 4ac = 0\;and\;a > 0$ , then domian of the function $f\left( x \right) = \log \left\{ {\left( {a{x^2} + bx + c} \right)\left( {x + 1} \right)} \right\}$ is

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x+y)=$ $f(x)+f(y)$ for all $x, y$ belonging to $\mathbb{R}$. If $m$ and $n$ are integers, then $f(m / n)$ is equal to

If $R$ denotes the set of all real numbers then the function $f: R \rightarrow R$ defined $f(x)=[x]$

If $|x-1| > 5$, then

Let $f\;:N \to N$ be defined by $f\left( x \right) = {x^2} + x + 1,$ then f is

If $A$ contains 10 elements then total number of functions defined from $A$ to $A$ is:

A function $f\left( x \right)$ is given by $f\left( x \right) = \frac{{{5^x}}}{{{5^x} + 5}}$, then the sum of the series $f\left( {\frac{1}{{20}}} \right) + f\left( {\frac{2}{{20}}} \right) + f\left( {\frac{3}{{20}}} \right) + ... + f\left( {\frac{{39}}{{20}}} \right)$ is equal to

Let $f:\left( { - 1,\;1} \right) \to B$, be a function defined by $f\left( x \right) = {\tan ^{ - 1}}\frac{{2x}}{{1 - {x^2}}}$, then $f$ is both one-one and onto when $B$ is the interval

Let $R$ be the set of real numbers and the mapping $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by $f(x)=5-x^2$ and $g(x)=3 x-4$, then the value of $(f o g)(-1)$ is