Question

The range of $f\left(x\right)=\frac{x^2-81}{x-9}$ is

• A. $R-\left\{18\right\}$
• B. $R-\left\{9\right\}$
• C. $R-\{\pm 9\}$
• D. $R-\{\pm 18\}$

Answer 1

The correct option is A $R-\left\{18\right\}$

$f\left(x\right)=\frac{x^2-81}{x-9}$
$f$ is defined for all $x$ except where $x-9=0$ i.e., $x=9$
$\therefore D\left(f\right)=R-\left\{9\right\}$

$f\left(x\right)=\frac{(x-9)(x+9)}{(x-9)},x\ne 9$
$\Rightarrow f\left(x\right)=x+9$
Now, $-\infty <x<\infty ,x\ne 9$
$\Rightarrow -\infty <x+9<\infty ,x\ne 9$
When $x=9,x+9=18$
$\therefore R\left(f\right)=R-\left\{18\right\}$