Question

Find the domain.the following function: √(x2−2x−2)+√(2x−3)=5

Answer 1

To find the domain of the given function, we need to consider the values of x that make the function defined. The domain of the function is x ≥ 1 + √3.

Step-by-step explanation:

To find the domain of the given function, we need to consider the values of x that make the function defined. The given function is a combination of two square roots, so we need to ensure that both square roots have non-negative arguments. We can start by considering the first square root:
`√(x^2 - 2x - 2).` The argument of this square root must be greater than or equal to 0. We can solve this inequality by finding the zeros of the quadratic expression:
`x^2 - 2x - 2 = 0.` Using the quadratic formula, we find x = 1 ± √3. Therefore, the first square root is defined for x values greater than or equal to 1 + √3 and less than or equal to 1 - √3.

Next, let's consider the second square root: √(2x - 3). The argument of this square root must also be greater than or equal to 0. Solving this inequality, we find 2x - 3 ≥ 0, which gives x ≥ 3/2. Therefore, the second square root is defined for x values greater than or equal to 3/2.

The domain of the given function is the intersection of the domains of the individual square roots. Therefore, x must satisfy both conditions: x ≥ 1 + √3 and x ≥ 3/2. Combining these inequalities, we find x ≥ 1 + √3. So, the domain of the function is x ≥ 1 + √3.