Lisantiom

2022-10-03

Determine the natural domain of $f\left(x\right)=\frac{\sqrt{N+3-{x}^{2}}}{\mathrm{log}\left(x\right)}$

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falwsay

Expert

For the square root to yield real numbers it must hold that
${x}^{2}\le N+3,$
meaning
$x\in \left[-\sqrt{N+3};\sqrt{N+3}\right].$
Now consider the denominator. First of all, the natural logarithm must take in positive values of x, so $x>0$. Besides that, the denominator cannot be 0. The logarithm assumes that value at $x=1$
Now what you do is the intersection of all of these constraints to obtain your domain.
$x\in \left[-\sqrt{N+3};\sqrt{N+3}\right]\cap \left(0;1\right)\cap \left(1;\mathrm{\infty }\right).$
Finally
$x\in \left(0;\sqrt{N+3}\right]\setminus \left\{1\right\}.$
If N is such that the square root stays below 1, the set substraction operation will just yield the interval $\left(0;\sqrt{N+3}\right]$

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