Find the Domain 4x square root of 2x cube root of 3x

Problem

4*x√(,2*x√(3,3*x))

Solution

1. Identify the constraints for the square root function. For the expression to be defined in the set of real numbers, the radicand (the expression inside the square root) must be greater than or equal to zero.

2*x√(3,3*x)≥0

2. Analyze the cube root function. The cube root √(3,3*x) is defined for all real numbers, so it does not impose any additional restrictions on the domain.

3. Solve the inequality for the square root radicand. Since the constant 2 is positive, we focus on the product of x and the cube root.

x√(3,3*x)≥0

4. Determine the sign of the product. If x>0 then 3*x>0 which means √(3,3*x)>0 The product of two positive numbers is positive.

x⋅(positive)>0

5. Check the case where x<0 If x<0 then 3*x<0 which means √(3,3*x)<0 The product of two negative numbers is positive.

x⋅(negative)>0

6. Check the case where x=0 If x=0 the expression evaluates to zero, which is allowed under a square root.

0⋅√(3,0)=0

7. Conclude that the expression inside the square root is always non-negative for any real number x Therefore, there are no restrictions on x

Final Answer

Domain:(−∞,∞)

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