Determine the natural domain of f ( x ) = N + 3 − x 2 log ⁡ ( x )

Question

Determine the natural domain of   f  (  x  )  =                    N        +        3        −                  x          2                            log        ⁡        (        x        )

falwsay · Accepted Answer

For the square root to yield real numbers it must hold that

x^{2} \leq N + 3,

meaning

x \in [- \sqrt{N + 3}; \sqrt{N + 3}] .

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 [- \sqrt{N + 3}; \sqrt{N + 3}] \cap (0; 1) \cap (1; \infty) .

Finally

x \in (0; \sqrt{N + 3}] ∖ {1} .

If N is such that the square root stays below 1, the set substraction operation will just yield the interval

(0; \sqrt{N + 3}]

Determine the natural domain of function f given by f(x)=(sqrt(N+3−x^2))/(log(x))

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