Question

The range of the function $f(x)=\frac{e^x\ln x{5}^{\left(x^2+2\right)}\left(x^2-7x+10\right)}{2x^2-11x+12}$ is

• A. $(-\infty ,\infty )$
• B. $[0,\infty )$
• C. $\left(\frac{3}{2},\infty \right)$
• D. $\left(\frac{3}{2},4\right)$

Answer 1

Answer: (A)

$f(x)=\frac{e^x\ln x{5}^{\left(x^2+2\right)}\cdot (x-2)(x-5)}{(2x-3)(x-4)}$
Note that at $x=3/2$ and $x=4$ function is not defined and in open interval $(3/2,4)$ function is continuous.
$\underset{x\to {\frac{3}{2}}^{+}}{\mathrm{Lim}}=\frac{(+ve)(-ve)(-ve)}{(+ve)(-ve)}\to -\infty$
$\underset{x\to 4^{-}}{\mathrm{Lim}}=\frac{(+ve)(+ve)(-ve)}{(+ve)(-ve)}\to \infty$
In the open interval $(3/2,4)$ the function is continuous and takes up all real values from $(-\infty ,\infty )$ Hence range of the function is $(-\infty ,\infty )$ or $R$