Question

Find the domain of the following function:

If $f\left(x\right)=\sqrt{x^2-5x+4}$ and $g\left(x\right)=(x+3)$, then find the domain of $\frac{f}{g}\left(x\right)$

• A. $(-\infty ,1]\cup [4,\infty )$
• B. $(-\infty ,-3)\cup (-3,1)\cup (3,\infty )$
• C. $(-\infty ,-3)\cup (-3,1)\cup (4,\infty )$
• D. None of these

Answer 1

Answer: (C)

$(-\infty ,-3)\cup (-3,1)\cup (4,\infty )$

$\frac{f}{g}\left(x\right)=\frac{\sqrt{x^2-5x+4}}{x+3}$

For $\frac{f}{g}$ to be defined,

$\Rightarrow x^2-5x+4\ge 0$

$(x-4)(x-1)\ge 0$

$\Rightarrow x\in (-\infty ,1]\cup [4,\infty )$

Also $x+3\ne 0$

$\Rightarrow x\ne -3$

So $x\in (-\infty ,-3)\cup (-3,1)\cup (4,\infty )$