f : R $\to$ R, then f(x) = $\mathrm{x}\left|\mathrm{x}\right|$ will be

• many-one-onto

• one-one-onto

• many-one-into

• one-one-into

The inverse ofthe function y = $\frac{2^{\mathrm{x}}}{1 + 2^{\mathrm{x}}}$ is

• $\mathrm{x} = {\log }_2\left(\frac{1}{1 - 2^{\mathrm{y}}}\right)$

• $\mathrm{x} = {\log }_2\left(1 - \frac{1}{\mathrm{y}}\right)$

• $\mathrm{x} = {\log }_2\left(\frac{1}{1 - \mathrm{y}}\right)$

• $\mathrm{x} = {\log }_2\left(\frac{\mathrm{y}}{1 - \mathrm{y}}\right)$

The domain of the definition of the function

$\mathrm{y} = \frac{1}{{\log }_{10}\left(1 - \mathrm{x}\right)} + \sqrt{\left(\mathrm{x} +\hspace{0.17em}2\right)}$ is

• $\mathrm{x} \ge - 2$

• $- 3 < \mathrm{x} \le - 2$

• $- 2 \le \mathrm{x} < 0$

• $- 2 \le \mathrm{x} < 1$

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

• many-one onto function

• many-one into function

• one-one onto function

• one-one into function

If domain of the function f(x) = x² - 6x + 7 is ($- \infty , \infty$), then its range is

• [ - 2, 3]

• $\left(- \infty , 2\right)$

• $\left(- \infty , \infty \right)$

• $[ - 2, \infty )$

Let R = {(1, 1), (1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A = {1, 2, 3, 4}. The relation R is

• a function

• transitive

• not symmetric

• reflexive

The relation R in R defined by R = {(a, b): a $\le$ b³), is

• reflexive

• symmetric

• transitive

• None of these

The value of the expression 1 . (2 - w)(2 - w²) + 2 . (3 - w)(3 - w²) + ... + (n - 1)(n - w²), where w is an imaginary cube root of unity is

• ${\left\{\frac{\mathrm{n}\left(\mathrm{n} + 1\right)}{2}\right\}}^2$

• ${\left\{\frac{\mathrm{n}\left(\mathrm{n} + 1\right)}{2}\right\}}^2 - \mathrm{n}$

• ${\left\{\frac{\mathrm{n}\left(\mathrm{n} + 1\right)}{2}\right\}}^2 + \mathrm{n}$

• None of these

Inverse of function f(x) = $\frac{{10}^{\mathrm{x}} - {10}^{- \mathrm{x}}}{{10}^{\mathrm{x}} + {10}^{- \mathrm{x}}}$ is

• log₁₀(2 - x)

• $\frac{1}{2}{\log }_{10}\left(\frac{1 + \mathrm{x}}{1 - \mathrm{x}}\right)$

• $\frac{1}{2}{\log }_{10}\left(2\mathrm{x} - 1\right)$

• $\frac{1}{4}{\log }_{10}\left(\frac{2\mathrm{x}}{2 - \mathrm{x}}\right)$

Let f : R - {x} $\to$ R be a function defined by f(x) =  $\frac{\mathrm{x} - \mathrm{m}}{\mathrm{x} - \mathrm{n}}$, where m $\ne$ n. Then

• f is one-one onto

• f is one-one into

• f is many one onto

• f is many one into