Question

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

• A. $[2,\infty )$
• B. $(2,\infty )$
• C. $[2,6)\cup (6,\infty )$
• D. $(-1,\infty )$

Answer 1

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

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


The graph of $y=f\left(x\right)$ is shown in figure, that shows $f\left(1\right)$ is included from left branch corresponding to $f(-3).$
$f_{min}$ occurs at $x=\frac{-2}{2}=-1$
$f_{min}=2$
Hence, its range is $[2,\infty )$