Answer:
In order to find the domain of the inequality g(x) ≥ -2√((1/2)x²), we need to determine the values of x that make the inequality true. In this case, there are no restrictions on the domain, so x can be any real number.
Explanation:
To find the domain of the inequality g(x) ≥ -2√((1/2)x², we need to consider two conditions:
- 1. The expression under the square root should be non-negative.
- 2. The denominator of the fraction, (1/2)x², should not equal zero.
Let's break it down step-by-step:
1. Non-Negative Expression:
To ensure the expression under the square root is non-negative, we set (1/2)x² ≥ 0.
To find the values of x that satisfy this inequality, we can consider two cases:
- a) When (1/2)x² > 0: In this case, x can be any non-zero real number, since any non-zero number squared is always positive.
- b) When (1/2)x² = 0: In this case, x must be equal to zero to satisfy the inequality.
Therefore, the values of x that satisfy the non-negative expression are all real numbers except x = 0.
2. Denominator ≠ 0:
To find the values of x that make the denominator (1/2)x² ≠ 0, we need to solve (1/2)x² ≠ 0.
To determine the values that satisfy this inequality, we can consider two cases:
- a) When x ≠ 0: In this case, the denominator (1/2)x² is always non-zero.
- b) When x = 0: In this case, the denominator becomes (1/2)(0)² = 0. However, since x = 0 satisfies the non-negative expression from step 1, we can include it in the domain.
Combining both conditions, we find that the domain of the inequality g(x) ≥ -2√((1/2)x² is all real numbers, including x = 0.
In summary, the domain of g(x) is the set of all real numbers.