Question

The range of $f\left(x\right)=\frac{e^x}{1+\left[x\right]},x\ge 0$ is

• A. $[1,\infty )$
• B. $R$
• C. $[0,\infty )$
• D. $R-[-1,0)$

Answer 1

The correct option is A $[1,\infty )$

$f\left(x\right)=\frac{e^x}{1+\left[x\right]},x\ge 0$
Clearly, $f\left(x\right)>0$ since numerator and denominator both are positive for $x\ge 0$.
As we increase the value of $x$, the numerator increases at much higher pace than the denominator.
So, maximum value of the function tends to $\infty$.

Now, $f\left(0\right)=\frac{e^0}{1+\left[0\right]}=\frac{1}{1+0}=1$
From the options, we can conclude that $R\left(f\right)=[1,\infty )$