 # Domain of One Variable Function – A function with square root – Exercise 5738

Exercise

Determine the domain of the function:

$$y=\sqrt{3x-x^3}$$

$$x\leq -\sqrt{3}\text{ or } 0\leq x\leq \sqrt{3}$$

Solution

Let’s find the domain of the function:

$$y=\sqrt{3x-x^3}$$

Because there is a square root, the expression inside the root must be non-negative:

$$3x-x^3\geq 0$$

Let’s find the roots of the polynomial equation:

$$3x-x^3=0$$

Factor the polynom:

$$x(3-x^2)=0$$

$$x(x-\sqrt{3})(x+\sqrt{3})=0$$

Therefore, its roots are

$$x=0,\pm\sqrt{3}$$

We are interested in the domain that holds this inequality:

$$x(x-\sqrt{3})(x+\sqrt{3})\geq 0$$

Therefore, the solution is

$$x\leq -\sqrt{3}\text{ or } 0\leq x\leq \sqrt{3}$$

The polynom:

$$y=3x-x^3$$

looks like this: The domain is marked in green lines.

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