The inequation `sqrt(f(x)) gt g(x)`, is equivalent to the

Options

A. `g(x) le 0 " " &" "f(x) ge 0" "or" "g(x) ge 0" "&" "f(x) gt g(x)`
B. `g(x) le 0 " " &" "f(x) ge 0" "or" "g(x) ge 0" "&" "f(x) gt g^(2)(x)`
C. `g(x) le 0 " " &" "f(x) ge 0" "or" "g(x) ge 0" "&" "f^(2)(x) gt g^(2)(x)`
D. `g(x) le 0 " " &" "f(x) ge 0" "or" "g(x) ge 0" "&" "f(x) lt g(x)`

Answer

To solve the inequation \( \sqrt{f(x)} > g(x) \), we can follow these steps:

### Step 1: Start with the given inequation
We begin with the inequation:
\[ \sqrt{f(x)} > g(x) \]

### Step 2: Square both sides
To eliminate the square root, we square both sides of the inequation. However, we must ensure that both sides are non-negative for the squaring to be valid. Therefore, we require \( f(x) \geq 0 \) and \( g(x) \) must be less than or equal to zero:
\[ f(x) > g(x)^2 \]

### Step 3: Consider the conditions for squaring
Since we squared both sides, we must also consider the conditions under which this operation is valid. We have:
1. \( f(x) \geq 0 \) (since we took the square root)
2. \( g(x) \) can be any real number, but for the inequality to hold after squaring, we need to ensure that \( g(x) \) is less than or equal to zero.

### Step 4: Combine the conditions
Thus, we combine our findings:
\[ f(x) > g(x)^2 \quad \text{and} \quad g(x) \leq 0 \]

### Conclusion
The inequation \( \sqrt{f(x)} > g(x) \) is equivalent to:
\[ f(x) > g(x)^2 \quad \text{and} \quad g(x) \leq 0 \]