Question

If $f\left(x\right)=x+5$ and $g\left(x\right)=\sqrt{x^2-9}$, then find the domain of gof (x)

• A. $(-8,-2)$
• B. ($\infty ,-8)\cup (-2,\infty$ )
• C. (-$\infty ,-8]\cup [-2,\infty$ )
• D. (- ($\infty ,-8]\cup [-2,\infty$ )

Answer 1

The correct option is C (-$\infty ,-8]\cup [-2,\infty$ )

We have, $f\left(x\right)=x+5$ and $g\left(x\right)=\sqrt{x^2-9}$

Now, $g\left(f\right(x\left)\right)$

$=g(x+5)$

$=\sqrt{(x+5{)}^2-9}$

$=\sqrt{x^2+25+10x-9}$

$=\sqrt{x^2+10x+16}$

$=\sqrt{(x+8)(x+2)}$

Clearly, $g\left(f\right(x\left)\right)$ is defined for

$(x+8)(x+2)\ge 0$

$\Rightarrow x\le -8orx\ge -2$

$\Rightarrow (-\infty ,-8]\cup [-2,\infty )$

Hence, $\text{C}$ is the correct option.