F : R to R is defined as $f(x) = left{ {begin{array}{*{20}{c}} {{x^2} + 2mx - 1,,}&{x ≤ 0

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1.Relation and Function

normal

$f : R \to R$ 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

A

$( - \infty ,0)$

B

$\left( { - \infty ,0} \right]$

C

$\left( {0,\infty } \right)$

D

$\left[ {0,\infty } \right)$

Solution

For $f$ to be one – one vertex must lie on or to the right of $y -$ axis.

$\therefore – m \ge 0 \Rightarrow m \le 0$

for $m = 0,\,\,f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
{{x^2} – 1,}&{x \le 0}\\
{ – 1,}&{x > 0}
\end{array}} \right.$ which is not

one – one

$\therefore m \in \left( { – \infty ,0} \right)$

Standard 12

Mathematics

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