Question

Range of the expression $f\left(x\right)=\left(\frac{x^3+x^2+x-3}{x-1}\right)$ is

• A. $[2,\infty )$
• B. $(2,\infty )$
• C. $[2,\infty )-\left\{6\right\}$
• D. $(-1,\infty )$

Answer 1

The correct option is A $[2,\infty )$

Given,

$f\left(x\right)=\left(\frac{x^3+x^2+x-3}{x-1}\right)$

Now,

$f\left(x\right)=\frac{x^3+x^2+x-3}{x-1}=\frac{(x^3-1)+(x^2-1)+(x-1)}{x-1}$
$\Rightarrow f\left(x\right)=\frac{(x-1)(x^2+x+1)+(x-1)(x+1)+(x-1)}{(x-1).1}$
$\Rightarrow f\left(x\right)=x^2+x+1+x+1+1$

$=x^2+2x+3$

$=(x+1{)}^2+2$
Hence range of $f\left(x\right)$ is $[2,\infty )$