Question

Let $f\left(x\right)=\ln x$ and $g\left(x\right)=\left(\frac{x^4-x^3+3x^2-2x+2}{2x^2-2x+3)}\right)$. The domain of $f\left(g\right(x\left)\right)$ is

• A. $(-\infty ,\infty )$
• B. $[0,\infty )$
• C. $(0,\infty )$
• D. $[1,\infty )$

Answer 1

The correct option is A $(-\infty ,\infty )$

$f\left(x\right)=\ln x$ and $g\left(x\right)=\frac{x^4-x^3+3x^2-2x+2}{2x^2-2x+3}$
Now, $f\left(g\right(x\left)\right)=f\left(\frac{x^4-x^3+3x^2-2x+2}{2x^2-2x+3}\right)$
$\Rightarrow f\left(g\right(x\left)\right)=\ln \left(\frac{x^4-x^3+3x^2-2x+2}{2x^2-2x+3}\right)$
For log to be defined
$\frac{x^4-x^3+3x^2-2x+2}{2x^2-2x+3}>0$
The above inequality holds for all $x\in R$