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Question:

Let f(x) = x², x ∈ R. For any A ⊆ R, define g(A) = {x ∈ R | f(x) ∈ A}. If S = [0, 4], then which one of the following statements is not true?

f(g(S)) = S

f(g(S)) ≠ f(S)

g(f(S)) = f(S)

g(f(S)) ≠ S

Solution:

The correct option is C g(f(S)) = f(S)
g(S) = [-2, 2]
So, f(g(S)) = [0, 4] = S
And f(S) = [0, 16] ⇒ f(g(S)) ≠ f(S)
Also, g(f(S)) = [-4, 4] ≠ g(S)
So, g(f(S)) ≠ S