Question

$f:(0,\infty )\to (0,\infty )$ defined by
$f\left(x\right)=\{\begin{array}{cc}{2}^x& x\in (0,1)\\ 5^x& x\in [1,\infty )\end{array}$ 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

$f\left(x\right)=2^x,x\in (0,1);$
$=5^x,x\in [1,\infty ]$
Range of $f\left(x\right)$ is $(1,2)\cup [5,\infty )$ that is not equal to its codomain.
But $f\left(x\right)$ is monotonic increasing function.
So $f\left(x\right)$ is one - one but not onto.