Question

Range of the function $f\left(x\right)=\ln {\left(\right\{x\}}^2+3\left\{x\right\}+2)$ is (where $\{.\}$ is fractional part function)

• A. $[2,6)$
• B. $(\ln 2,\ln 6)$
• C. $[\ln 2,\ln 6)$
• D. $(-\infty ,\ln 2]$

Answer 1

Answer: (C)

$[\ln 2,\ln 6)$

$\left\{x^2\right\}+3\left\{x\right\}+2={(\left\{x\right\}+\frac{3}{2})}^2+2-\frac{9}{4}$

$={(\left\{x\right\}+\frac{3}{2})}^2-\frac{1}{4}$

$0\le \left\{x\right\}<1$

$\therefore$ Range of ${(\left\{x\right\}+\frac{3}{2})}^2-\frac{1}{4}$ is $[2,6)$

Range of $f\left(x\right)$ is $[\ln 2,\ln 6)$