Functions practice · Mixed difficulty

The domain of definition of

$f\left( x \right) = {\log _{10}}\left\{ {{{\log }_{10}}\left( {1 + {x^3}} \right)} \right\},\;\mathrm{is}$

Let $f:R \to R,g:R \to R$ be two functions given by $f\left( x \right) = 2x - 3,g\left( x \right) = {x^3} + 5.$ Then, ${\left( {fog} \right)^{ - 1}}x$ is equal to

The solution set of the inequality $\mathrm{lo}{\mathrm{g}_{\mathrm{sin}\left( {\frac{\pi }{3}} \right)}}\left( {{x^2} - 3x + 2} \right) \ge 2$ is

Let $f: \mathrm{A} \rightarrow \mathrm{B}$ be a function defined as $f(x)=\frac{x-1}{x-2}$, where $A=R-\{2\}$ and $B=R-\{1\}$. Then $f$ is

Let $R$ be a relation in $N$ defined by $R=\left\{\left(1+x, 1+x^2\right): x \leq 5, x \in N\right\}$.
Which of the following is false?

The domain of the real function $f\left( x \right) = \frac{1}{{\sqrt {4 - {x^2}} }}$ is

Let $f: R \rightarrow R$ be a function defined by $f(x)=x^3+4$, then $f$ is

Suppose that $f$ is continuous on $[\mathrm{a}, \mathrm{b}]$ and that $f(\mathrm{x})$ is an integer for each $\mathrm{x}$ in $[\mathrm{a}, \mathrm{b}]$. Then in $[\mathrm{a}, \mathrm{b}]$

Let $A = \left\{ {1,2,3} \right\}\;and\;B = \left\{ {2,3,4} \right\}$ then which of the following relations is a function from $A\;to\;B?$

$f\;:\left[ { - 4,0} \right] \to R$ is given by$f\left( x \right) = {e^x} + \sin x,$ its even extension to $\left[ { - 4,4} \right],$ is

Let $A \equiv\{1,2,3,4\}, B \equiv\{a, b, c\}$, then number of functions from $A \rightarrow B$, which are not onto i

If R denotes the set of all real numbers, then the function $f:R \to R$ defined by $f\left( x \right) = \left| x \right|$ is

If $f\left( x \right) = \sqrt {\cos \left( {\sin x} \right)} + \sqrt {\sin \left( {\cos x} \right)} ,$ then range of $f\left( x \right)$ is

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

If sides of $\triangle A B C$ are $a, b$ and $c$ such that $2 b=a+c$, then exhaustive range of $\frac{b}{c}$ is

Find the number of integers not in the domain of $f(x)=\cos ^{-1}\left(\frac{2-x}{2 x}\right)$.

Solution set of inequality ${\log _e}\frac{{x - 2}}{{x - 3}}$ is

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

If $f: \mathbb{R} \rightarrow \mathbb{R} \& \quad g: \mathbb{R} \rightarrow \mathbb{R}$ be two given functions, then $2 \min \{f(x)-g(x), 0\}$ equals

Which of the following statement is true $(\mathrm{T})$ or false $(\mathrm{F})$ for the function $f: \mathrm{R} \rightarrow \mathrm{R}$ ?
(i) If $|f(x)-f(y)| \leq 30|x-y|, \forall x, y \in \mathrm{R}$, then $f$ is a continuous function on $\mathrm{R}$.
(ii) If $|f(x)-f(y)| \leq 30|x-y|, \forall x, y \in \mathrm{R}$, then $f$ is differentiable on $\mathrm{R}$.
(iii) If $|f(x)-f(y)| \leq 21|x-y|^2, \forall x, y \in \mathbb{R}$, then $f$ is differentiable on $\mathrm{R}$.
(iv) If $|f(x)-f(y)| \leq 21|x-y|^2, \forall x, y \in \mathrm{R}$, then $f$ is a constant function.

The set of all real $x$ satisfying the inequality $\frac{{3 - \left| x \right|}}{{4 - \left| x \right|}} > 0$

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

If $f(x)=a \cos (b x+c)+d$, then range of $f(x)$ is

If $f(x)$ is an odd function, $f(1)=3$, and $f(x+2)$ $=f(x)+f(2)$, then the value of $f(3)$ is

If from $R \rightarrow R, f(x)=(x+1)^2, g(x)=x^2+1$, then $($ fog $)(-3)$ equals

If $f(x)=\{x\}+\{x+1\}+\{x+2\} \ldots \ldots . . x+99\}$, then the value of $[f(\sqrt{2})]$ is, where $\{$.$\}$ denotes fractional part function & $[.]$ denotes the greatest integer function

If $f:R \to R\;and\,g:R \to R$ are defined by $f\left( x \right) = x - \left[ x \right]\;and\;g\left( x \right) = \left[ x \right]$ for $x \in R$, where $\left[ x \right]$ is the greatest integer not exceeding x, then for every $x \in R,\;f\left( {g\left( x \right)} \right)$ is equal to

If $f: R \rightarrow R$ is given by $f(x)=3 x-5$, then $f^{-1}(x)$ ?

Suppose that $g(x)=1+\sqrt{x}$ and $f(g(x))=3+2 \sqrt{x}+x$, then $f(x)$ is:

Let $f$ be a function satisfying $2 f(x)-3 f(1 / x)=x^2$ for any $x \neq 0$. Then the value of $f(2)$ is