Let `f(x)=x^(2), x in R. " for any " A subset eq R,` define `g(A)={x in R: f(x) in A}`. If ` S=[0,4],` then which one of the following statements is not true?
(A) `f(g(S))=S`
(B) `g(f(S)) ne S`
(C) `g(f(S)) =g(S)`
(D) `f(g(S))ne f(S)`

Options

A. `f(g(S))=S`
B. `g(f(S)) ne S`
C. `g(f(S)) =g(S)`
D. `f(g(S))ne f(S)`

Explanation

To solve the problem, we need to analyze the functions and sets given in the question.

1. **Understanding the Functions**:
- We have \( f(x) = x^2 \) where \( x \in \mathbb{R} \).
- We define \( g(A) = \{ x \in \mathbb{R} : f(x) \in A \} \).

2. **Given Set**:
- We are given \( S = [0, 4] \).

3. **Finding \( g(S) \)**:
- We need to find \( g(S) \).
- Since \( f(x) = x^2 \), we need to find all \( x \) such that \( f(x) \in [0, 4] \).
- This means \( 0 \leq x^2 \leq 4 \).
- Taking square roots, we find \( -2 \leq x \leq 2 \).
- Therefore, \( g(S) = [-2, 2] \).

4. **Finding \( f(g(S)) \)**:
- Now we compute \( f(g(S)) = f([-2, 2]) \).
- The range of \( f(x) \) for \( x \in [-2, 2] \) is \( [0, 4] \) since \( f(-2) = 4 \) and \( f(2) = 4 \), and \( f(0) = 0 \).
- Thus, \( f(g(S)) = [0, 4] \).

5. **Finding \( f(S) \)**:
- Next, we find \( f(S) = f([0, 4]) \).
- The range of \( f(x) \) for \( x \in [0, 4] \) is \( [0, 16] \) since \( f(0) = 0 \) and \( f(4) = 16 \).
- Therefore, \( f(S) = [0, 16] \).

6. **Finding \( g(f(S)) \)**:
- Now we compute \( g(f(S)) = g([0, 16]) \).
- We need to find all \( x \) such that \( f(x) \in [0, 16] \).
- This means \( 0 \leq x^2 \leq 16 \).
- Taking square roots, we find \( -4 \leq x \leq 4 \).
- Therefore, \( g(f(S)) = [-4, 4] \).

7. **Comparing Results**:
- We have:
- \( f(g(S)) = [0, 4] \)
- \( g(f(S)) = [-4, 4] \)
- Now we check the options provided in the question to find which statement is not true.

8. **Checking the Options**:
- Option A: \( f(g(S)) = S \) → True, since both are \( [0, 4] \).
- Option B: \( g(f(S)) = [-4, 4] \) → True.
- Option C: \( g(f(S)) = g(S) \) → False, since \( g(S) = [-2, 2] \) and \( g(f(S)) = [-4, 4] \).
- Option D: \( f(g(S)) \neq f(S) \) → True, since \( [0, 4] \neq [0, 16] \).

Thus, the statement that is not true is option C.