Final answer:
The domain of the function g(x) is found by ensuring the expression under the square root is non-negative. After solving the inequality, we find the domain of g(x) is all real numbers x that satisfy x ≤ -4.
Step-by-step explanation:
To find the domain of the inequality g(x) ≥ -2√(1/2)x+2, we must consider the expression under the square root, because the square root function is only defined for non-negative numbers. The expression inside the square root must be greater than or equal to zero.
In this case, the inequality becomes 1/2x + 2 ≥ 0, which simplifies to x ≥ -4. However, we must also consider the multiplication by -2. Since we are looking for g(x) ≥ 0, and the square root expression is multiplied by -2, we need to ensure that the square root expression is ≤ 0, so our initial inequality changes to 1/2x + 2 ≤ 0.
Now, we solve for x to find the domain:
- Multiply both sides of the inequality by 2: x + 4 ≤ 0.
- Subtract 4 from both sides: x ≤ -4.
Therefore, the domain of the function g(x) is all real numbers x that satisfy x ≤ -4.