Functions \( f(x) \) and \( g(x) \) are both increasing. In addition, the range of \( g \) is in the domain of \( f \). Prove that the composite function \( f(g(x)) \) is increasing.
Answer (2)
f: D -> E; g: F -> G, where, g(F) is in D;
Let be x1, x2 ∈ F, with x1 < x2; But, g(x1) < g(x2), because g is increasing; => f(g(x1) < f(g(x2) , because f is increasing => f(g(x) is increasing.
Given that both functions f and g are increasing and the range of g is in the domain of f , we can conclude that the composition f ( g ( x )) is also increasing. This is proven by showing that for any two inputs x 1 and x 2 where x 1 < x 2 , it follows that f ( g ( x 1 )) < f ( g ( x 2 )) . Therefore, f ( g ( x )) is increasing.
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