Question

$f:R\to R$ is defined as $f\left(x\right)=3^x$ then which of the following is correct
I) $f$ is one-one
II) $f$ is onto
III) $f$ is a increasing function

• A. $I$, $II$ only
• B. $II$, $III$ only
• C. $I$, $III$ only
• D. $I$, $II$, $III$

Answer 1

Answer: (C)

$I$, $III$ only

$f\left(x_1\right)=f\left(x_2\right)\Rightarrow 3^{x_1}=3^{x_2}$
$\log 3^{x_1}=\log 3^{x_2}$
$\Rightarrow x_1=x_2$
$\Rightarrow f\left(x_1\right)=f\left(x_2\right)\iff x_1=x_2$
$\therefore$ f is one-one
$f^1\left(x\right)=3^x{\log }_e^3>0$ for all x
hence f is increasing.
$f\left(x\right)>0$ for all x, hence $f:R\to R$ is not
on to as range $=R^{+}$ is not equal to co-domain R