Question

$f:(-\frac{\pi }{2},\frac{\pi }{2})\to (-\infty ,\infty )$ defined by $f\left(x\right)=1+3x$ is

• A. one-one but not onto
• B. onto but not one-one
• C. neither one - one nor onto
• D. bijective

Answer 1

The correct option is A one-one but not onto

$f\left(x_1\right)=f\left(x_2\right)\Rightarrow 1+3x_1=1+3x$
$x_1=x_2$
$\therefore f\left(x_1\right)=f\left(x_2\right)\Rightarrow x_1=x_2$
$\therefore$ f is one-one.
$f$ lies b/w $(1-\frac{3\pi }{2},\frac{1+3\pi }{2})$
$\therefore$ co-domain $(-\infty ,\infty )$ is not equal to range $(\frac{1-3\pi }{2},\frac{1+3\pi }{2})$
$\Rightarrow$ f is not onto.
$\therefore$ f is one-one but not onto.