Functions practice · Easy

Let $f:R - \{ 3\} \to R - \{ 1\}$ be defined by $f(x) = \frac{{x - 2}}{{x - 3}}$. Let $g:R \to R$ be given as $g(x) = 2x - 3$. Then, the sum of all the values of $x$ for which ${f^{ - 1}}(x) + {g^{ - 1}}(x) = \frac{{13}}{2}$ is equal to

If $f:(0, \pi) \rightarrow \mathbf{R}$ is given by $f(x)=\sum\limits_{{k=1}}^n[1+\sin k x],[x]$ denotes the greatest integer function, then the range of $f(x)$ is

The range of $f(x)=\cos x-\sin x$ is

If $f(x)=\frac{2 x+1}{3 x-2}$, then $(f \circ f)(2)$ is equal to

Let the function $f$ be defined by $f(x)=\frac{2 x+1}{1-3 x}$, then $f^{-1}(x)$ is

If $f\left( x \right) = x - \frac{1}{x},x \ne 0,$ then $f\left( {{x^2}} \right)$ equals

If $f\left( x \right)$ is an odd periodic function with period 2, then $f\left( 4 \right)$ equals

The domain of the function
$f\left( x \right) = \frac{{{{\sin }^1}\left( {x - 3} \right)}}{{\sqrt {9 - {x^2}} }}$ is

Let $n\left( A \right) = 4\,and\,n\left( B \right) = 6$. The number of one to one functions from A to B is

If a function F is such that $F\left( 0 \right) = 2,\;F\left( 1 \right) = 3,\;F\left( {n + 2} \right) = 2F\left( n \right) - F\left( {n + 1} \right)\;for\;n \ne 0$, then $F\left( 5 \right)$ is equal to

If $g\left( x \right) = 1 + \sqrt x \;and\;f\left( {g\left( x \right)} \right) = 3 + 2\sqrt x + x$ then, $f\left( x \right)$ is equal to

If $a f(x+1)+b f\left(\frac{1}{x+1}\right)=x, x \neq-1, a \neq b$, then $f(2)$ is equal to

Assertion: The function $f: R \rightarrow R$ given by $f(x)=x^3$ is injective.

Reason: The function $f: X \rightarrow Y$ is injective, if $f(x)=f(y)$ $\Rightarrow {x}={y}$ for all ${x},{y} \in {X}$.

If $3 < 3t - 18 \le 18$, then which one of the following is true?

If $f(x)=3 x-5$, then $f^{-1}(x)$

Let $f(x)$ is a monic polynomial of degree $=5$ such that $f(1)=1, f(2)=2, f(3)=3$, $f(4)=4$ and $f(5)=5$. If $f(6)=5 !+\lambda$, then find the value of $\lambda$.

If $f\;:R \to R$ is defined by $f\left( x \right) = 2x - 2\left[ x \right]$ for all $x \in R,$ where $\left[ x \right]$ denotes the greatest integer less than or equal to $x$, then range of $f$, is

If $f(x)=\frac{1}{(1-x)}, g(x)=f\{f(x)\}$ and $h(x)=f[f\{f(x)\}]$. Then the value of $f(x) \cdot g(x) \cdot h(x)$ is

The maximum value attained by the function $y=10-|x-10|$, if $-9 \leq x \leq 9$, is

The domain of definition of $f\left( x \right) = {\log _{1.7}}{\left( {\frac{{2 - \phi '\left( x \right)}}{{x + 1}}} \right)^{1/2}},$ where $\phi \left( x \right) = \frac{{{x^3}}}{3} - \frac{3}{2}{x^2} - 2x + \frac{3}{2},$ is

Let $f(n)=\left[\frac{1}{3}+\frac{3 n}{100}\right] n$, where $[\mathrm{n}]$ denotes the greatest integer less than or equal to $n$. Then $\sum_{n=1}^{56} f(n)$ is equal to:

The number of surjection from
$\mathrm{A}=\{1,2$...., $\mathrm{n}\}, \mathrm{n} \geq 2$ onto $\mathrm{B}=\{\mathrm{a}, \mathrm{b}\}$ is

Given $f\left( x \right) = \log \left( {\frac{{1 + x}}{{1 - x}}} \right)\;and\;g\left( x \right) = \frac{{3\;x + {x^3}}}{{1 + 3\;{x^2}}},$ then $fog\left( x \right)$ equals

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

The number of surjections from $A=\{1,2, \ldots, n\}, n \geq 2$ onto $B=\{a, b\}$ is

If $f(x)=\frac{a^x+a^{-x}}{2}$ and $f(x+y)+f(x-y)=K f(x)$ $f(y)$ then $K$ is equal to

Let ${f}({x})$ be a differentiable even function. Consider the following statements:

(I) ${f}^{\prime}({x})$ is an even function.

(II) ${f}^{\prime}({x})$ is an odd function.

(III) ${f}^{\prime}({x})$ may be even or odd.

Which of the above statements is/are correct?

If $f(x)=\cos \left[\pi^{2}\right] x+\cos \left[-\pi^{2}\right] x$, then

If $f:R \to R$ be a mapping defined by $f\left( x \right) = {x^3} + 5,$ then ${f^{ - 1}}\left( x \right)$ is equal to

Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be defined by $f(x)=\frac{|\mathrm{x}|-1}{|\mathrm{x}|+1}$ then $\mathrm{f}$ is: