Question

Find the domains of the following functions :

$f\left(x\right)=\sqrt{(5x-6-x^2)\left[\ln x\right]}+\sqrt{(7x-5-2x^2)}+{\left(\ln (\frac{7}{2}-x)\right)}^{-1}$

• A. $(1,2)\cup \left(2,\frac{5}{2}\right)$
• B. $(1,\frac{5}{2})$
• C. $(0,1)\cup \left(2,\frac{5}{2}\right)$
• D. none of these

Answer 1

Answer: (A)

$(1,2)\cup \left(2,\frac{5}{2}\right)$

For $\ln x$ to be defined $x\ne 1$

$\ln x\to (0,1)\Rightarrow \left[\ln x\right]=0$
& $2x^2-7x+5\le 0$

$\Rightarrow (2x-5)(x-1)\le 0\Rightarrow 1\le x\le \frac{5}{2}$

$\frac{1}{\ln \left(\frac{7}{2}-x\right)}$

$\frac{7}{2}-x>0\Rightarrow x<\frac{7}{2}$

also $\frac{7}{2}-x\ne 1\Rightarrow x\ne \frac{5}{2}$

$\therefore x\in (1,2)\cup (2,\frac{5}{2})$