Only positive or zero expressions are allowed under the root. So $x-36x-36$ must be positive or zero. $x-36x-36$ becomes zero for $x=36x\; =\; 36$.

• When we make $xx$ smaller (so ), then $x-36x-36$ gets negative (which is not good)

• When we make $xx$ larger (so $x>36x>36$), then $x-36x-36$ becomes positive (which is what we want).

So $x\ge 36x\; \backslash ge\; 36$ is the required condition for obtaining a positive or zero number under the root. Further, $xx$ is a real number, so

Only positive or zero expressions are allowed under the root. So $36+x^{2}36+x^2$ must not be negative. Now, $x^{2}x^2$ is never negative and if we add $+36+36$, it only becomes "more positive". So any $xx$ is allowed in the domain of definition, as long as it is a real number:

Only positive or zero expressions are allowed under the root. So $x+36x+36$ must not be negative. Further, we cannot divide by zero, which excludes the case $\sqrt{x+36}=0\backslash sqrt\{x+36\}\; =\; 0$, which happens if $x+36x+36$ becomes zero. So only $x+36>0x+36\; >\; 0$ is allowed. This is the case whenever $x>-36x>-36$. Further, we know that $xx$ is a real number.

So the domain of definition of $\frac{1}{\sqrt{\mathrm{x}+36}}\backslash frac1\{\backslash sqrt\{\backslash mathrm\; x+36\}\}$ is characterized by

Only positive or zero expressions are allowed under the root. So $x^2-36x^2-36$ must not be negative. That means, $x^{2}x^2$ must be larger or equal $3636$.

• For $x=6x\; =\; 6$ and $x=-6x\; =\; -6$, you exactly obtain $x^2=36x^2\; =\; 36$. This is a "good case"

• If $xx$ is positive, then making $xx$ larger (so $x>6x>6$) will make $x^{2}x^2$ larger than $3636$, which is also a "good case".

• Analogously, for negative $xx$, making $xx$ even more negative (so ) will also result in $x^2>36x^2\; >\; 36$, which is good for our purposes.

• However, if -, so $x\in (-6,6)x\; \backslash in\; (-6,\; 6)$, then $x^{2}x^2$ becomes smaller than 36, which is not good.. So we have to exclude this case.

Further, we know that $xx$ is a real number. So the domain of definition is given by