# Domain of One Variable Function – A function with fourth root in the denominator – Exercise 5732

Exercise

Determine the domain of the function:

$$y=\frac{1}{\sqrt[4]{4-x^2}}$$

$$|x|<2$$

Solution

Let’s find the domain of the function:

$$y=\frac{1}{\sqrt[4]{4-x^2}}$$

Because there is a denominator, the denominator must be different from zero:

$$\sqrt[4]{4-x^2}\neq 0$$

Also, there is a fourth root, so we need the expression inside the root to be non-negative:

$$4-x^2\geq 0$$

The two inequalities are equivalent to the inequality:

$$4-x^2>0$$

It is a square inequality. The roots of the quadratic equation:

$$4-x^2=0$$

are

$$x=\pm 2$$

Because we are looking for the section above the x-axis  and the parabola “cries”, we get that the solution of the inequality is

$$-2

By absolute value definition, this inequality is equivalent to

$$|x|<2$$

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