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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}}

Final Answer


|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<x<2

By absolute value definition, this inequality is equivalent to

|x|<2

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