Functions practice · Mixed difficulty

$f\left( x \right) = \left| {\sin x} \right|$ has an inverse if its domain is

The function $f(x)=\log \left(x+\sqrt{x^2+1}\right)$, is

The function $f: R \rightarrow R$ defined by $f(x)=e^x$ is:

Function $\sin ^{-1} \sqrt{x}$ is defined in the interval

Function $f: N \rightarrow N, f(x)=2 x+3$ is:

The value of $b$ and $c$ for which the identit $f(x+1)-f(x)=8 x+3$ is satisfied, wher $f(x)=b x^{2}+c x+d$, are

If $f\left( x \right) = {e^x}\;and\;g\left( x \right) = {\log _e}x,$ then which of the following is true?

Which of the following relation is NOT a function?

The range of the function $f(x)=\cos [x]$ for $-\pi / 2$ $< x < \pi / 2$ contains

If a function $f$ satisfies $f\left\{ {f\left( x \right)} \right\} = x + 1$ for all real values of $x$ and if $f\left( 0 \right) = \frac{1}{2}$, then $f\left( 1 \right)$ is equal to

The function $f: N \rightarrow N$ ($N$ is the set of natural numbers) defined by $f(n)=2 n+3$, is

If $f:R \to S$, defined by $f\left( x \right) = \sin x - \sqrt 3 \cos x - 1$ is onto, then the intervel of s is

Assertion: Let $\lim \limits_ {{x} \rightarrow {a}} {f}({x})=l$ and $\lim \limits_ {{x} \rightarrow {a}} {g}({x})={m}$. If $l$ and ${m}$ both exist, then $\lim \limits_{{x \rightarrow a}}(f g)(x)=\lim \limits_{{x \rightarrow a}} f(x). \lim \limits_{{x \rightarrow a}} g(x)=l m.$

Reason: Let ${f}$ be a real valued function defined by $f(x)=x^2+1$, then $f^{\prime}(2)=4$.

Set $A$ has 3 elements and set $B$ has 4 elements. The number of injection that can be defined from $A$ to $B$ is

The domain of the function $f(x)=\sqrt{x-x^{2}}+\sqrt{4+x}+\sqrt{4-x}$ is

The domain of definition of the function

$f\left( x \right) = x \cdot \frac{{1 + 2{{\left( {x + 4} \right)}^{ - 0.5}}}}{{2 - {{\left( {x + 4} \right)}^{0.5}}}} + {\left( {x + 4} \right)^{0.5}} + 4{\left( {x + 4} \right)^{0.5}}\mathrm{is}$

Let a function $f:N \to N$ be defied by $f\left( n \right) = \left[ {\begin{array}{cccccccccccccccccccc} {2n,}&{n = 2,4,6,8....} \\ {n - 1,}&{n = 3,7,11,15....} \\ {\frac{{n + 1}}{2},}&{n = 1,5,9,13....} \end{array}} \right.$ then $f$ is

If $f\left( x \right) = \left[ {x - 2} \right]$, where $\left[ x \right]$ denotes the greatest integer less than or equal to $x$, then $f\left( {2,\;5} \right)$ is equal to

The domain of definition of the function

$f\left( x \right) = {\log _3}\left\{ { - {{\log }_4}\left( {\frac{{6x - 4}}{{6x + 5}}} \right)} \right\},\;\mathrm{is}$

Let $f\left( x \right) = x + 1\;and\;\phi \left( x \right) = x - 2.$. Then the values of x satisfying $\left| {f\left( x \right) + \phi \left( x \right)} \right| = \left| {f\left( x \right)} \right| + \left| {\phi \left( x \right)} \right|$ are :

Let $f(x) = {a^x}(a > 0)$ be written as $f(x) = {f_1}(x) + {f_2}(x),$ where ${f_1}(x)$ is an even function and ${f_2}(x)$ is an odd function. Then ${f_1}(x + y) + {f_1}(x - y)$ equals

Which of the following functions is not an are not an injective map(s)?

Let $g\left( x \right) = 1 + x - \left[ x \right]$ and $f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {\begin{array}{cccccccccccccccccccc} { - 1,\;\;\;\;\;\;\;\;\;\;x < 0\;\;\;\;\;\;}\\ {0,\;\;\;\;\;\;\;\;\;\;\;\;\;x = 0,\;} \end{array}}\\ {1\;\;\;\;\;\;\;\;\;\;\;\;x > 0\;\;\;} \end{array}} \right.\;\mathrm{then}\;\mathrm{\;for}$ for $x,f\left[ {g\left( x \right)} \right]$ is equal to

The domain of the function $f(x)=\left(\log ^x\right)^{32}$ is

Let $f: N \rightarrow N$ where $N$ is set of natural numbers be a function such that $f(x+y)=f(x y) \, \forall \, x \geq 4$ and $y \geq 4$, then

Domain of $f(x)=\log |\log x|$ is

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

Let $\mathrm{A}$ and $\mathrm{B}$ be non-empty sets. Let $f: \mathrm{A} \rightarrow \mathrm{B}$ and $g: \mathrm{B} \rightarrow \mathrm{A}$ be two functions such that $f(g(x))=x, \forall x \in \mathbf{B}$. Which of the following statements are true (T) or false (F) ?
(i) The function $f$ is one-one function
(ii) The function $f$ is onto
(iii) The function $g$ is one-one
(iv) The function $g$ is onto

Let $f(x)=2^{10} \cdot x+1$ and $g(x)=3^{10} \cdot x-1$. If $(f \circ g)(x)=x$, then $\mathrm{x}$ is equal to :

Suppose $a,b$ and $c$ are real numbers such that $\frac{a}{b} > 1$ and $\frac{a}{c} < 0$. Which one of the following is true?