$f : R \to R$ is defined as

$f(x) = \left\{ {\begin{array}{*{20}{c}}
{{x^2} + 2mx – 1\,,}&{x \leq 0}\\
{mx – 1\,\,\,\,\,\,\,\,\,\,\,\,\,,}&{x > 0}
\end{array}} \right.$

 If $f (x)$ is one-one then the set of values of $'m'$ is

Let $A = \left\{ {{x_1},{x_2},{x_3},…..,{x_7}} \right\}$ and $B = \left\{ {{y_1},{y_2},{y_3}} \right\}$ be two sets containing seven and three distinct elements respectively. Then the total number of functions $f:A \to B$ which are onto, if there exist exactly three elements $x$ in $A$ such that $f(x) = {y_2}$ , is equal to

Show that the function $f: R \rightarrow R$ defined as $f(x)=x^{2},$ is neither one-one nor onto.

The function $f(x) = \;|px – q|\; + r|x|,\;x \in ( – \infty ,\;\infty )$, where $p > 0,\;q > 0,\;r > 0$ assumes its minimum value only at one point, if

Let $P(x)$ be a polynomial with real coefficients such that $P\left(\sin ^2 x\right)=P\left(\cos ^2 x\right)$ for all $x \in[0, \pi / 2)$. Consider the following statements:

$I.$ $P(x)$ is an even function.

$II.$ $P(x)$ can be expressed as a polynomial in $(2 x-1)^2$

$III.$ $P(x)$ is a polynomial of even degree.

Then,