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# Function

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Suppose that f(x) and g(x) are functions on R such that the range of f is [-5,3], and the range of g is [-2,1]. The range of f(x) + g(x) is [a,b]. What is the largest possible value of b?

Jun 16, 2022

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The maximum of f(x) + g(x) is always not greater than the maximum of f(x) + the maximum of g(x) because if the maximum point of f(x) and the maximum of g(x) point is not aligned on a vertical line, the maximum of f(x) + g(x) is going to be smaller.

That means, $$\max(f(x) + g(x)) = b \leq \max(f(x)) + \max(g(x)) = 3 + 1 = 4$$.

Then $$b \leq 4$$.

Jun 16, 2022