Question

If $f:R\to R$ defined by $f\left(x\right)=x,$ if $x>2;f\left(x\right)=5x-2$ if $x\le 2$ then $f$ is

• A. an identity function
• B. one one
• C. onto
• D. one one and onto

Answer 1

Answer: (D)

one one and onto

$f\left(x\right)=\{\begin{array}{cc}x& x>2\\ 5x-2& x⩽2\end{array}$
& $f^{\prime }\left(x\right)=\{\begin{array}{cc}1& x>2\\ 5& x⩽2\end{array}$ i.e. monotonic increasing.
And range of $f\left(x\right)$ is $(2,\infty )\cup (-\infty ,2]$
i.e. $(-\infty ,\infty )$
So $f\left(x\right)$ is both one - one & onto.