Exercise
Determine the domain of the function:
y=\sqrt{3x-x^3}
Final Answer
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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