Find the Domain x-3 square root of 3x^4-2x^3-x-1

Problem

ƒ(x)=(x−3)√(,3*x4−2*x3−x−1)

Solution

1. Identify the restriction for the domain. For a square root function 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.

3*x4−2*x3−x−1≥0

2. Analyze the polynomial P(x)=3*x4−2*x3−x−1 to find its roots. By testing small values, we find that x=1 is a root because 3*(1)4−2*(1)3−(1)−1=3−2−1−1=−1 which is not zero. Testing x=1.14… or using numerical methods reveals the real roots are approximately x≈−0.589 and x≈1.164

3. Determine the intervals where the polynomial is non-negative. Since the leading coefficient is positive (3>0, the quartic function opens upwards. The function is non-negative outside the interval between its two real roots.

4. Express the domain in interval notation based on the roots of the inequality. The expression x−3 is a polynomial and is defined for all real numbers, so it does not restrict the domain further.

Final Answer

Domain:(−∞,≈−0.589]∪[≈1.164,∞)

Want more problems? Check here!