Question

If $f\left(x\right)=\frac{\sqrt{x^2-4}}{x-4}$, find all the values of $x$ for which $f\left(x\right)$ is defined.

• A. All real numbers except $4$
• B. All real numbers except $-2$ and $2$
• C. All real numbers greater than or equal to $-2$ and less than or equal to $2$
• D. All real numbers less than or equal to $-2$ or greater than or equal to $2$
• E. All real numbers less than or equal to $-2$ or greater than or equal to $2$, except $4$

Answer 1

Answer: (E)

All real numbers less than or equal to $-2$ or greater than or equal to $2$, except $4$

Given, $f\left(x\right)=\frac{\sqrt{x^2-4}}{x+4}$

The denominator of function is equal to $0$ and numerator is equal to $0$.

If numerator is equal to $0$

Then $\sqrt{x^2-4}=0$

$\Rightarrow x^2-4=0$

$\Rightarrow x^2=4$

$\Rightarrow x=\pm 2$

If The denominator of function is equal to $0$, then $x-4=0$.

$\Rightarrow x=4$

Then all real numbers less than or equal to $-2$ or greater than or equal to $2$, except $4$.