Question

Find the range of the functions
i) $f\left(x\right)=\sqrt{x^2-x+1}$
ii) $f\left(x\right)=ln(x-[x\left]\right)$, where $[.]$ denotes the greatest integer function.

Answer 1

$\left(1\right)$ $f\left(x\right)=y=\sqrt{x^2-x+1}$

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

So, $min(x^2-x+1)=\frac{3}{4}$

Thus, range of $y$ is $[\frac{\sqrt{3}}{2},\infty )$

$\left(2\right)$ $f\left(x\right)=\ln (x-[x\left]\right)$

since $x-\left[x\right]=\left\{x\right\}$

$f\left(x\right)=\ln \left(\left\{x\right\}\right)$

as $\left\{x\right\}ϵ[0,1)$

so,

Thus, range of $f\left(x\right)$ is $(-\infty ,0)$