Let f(x) be a continuous function from R to R such that f(x) = 1 - f(2 - x). Which one of the follow

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Question

Let f(x) be a continuous function from R to R such that f(x) = 1 - f(2 - x). Which one of the following options is the CORRECT value of the integral from 0 to 2 of f(x) dx?

Accepted Answer

Correct Answer: (B) 1Concept -- Definite Integration and Functional Equations: Let I = integral from 0 to 2 of f(x) dx. Substitute x = 2 - t (so dx = -dt; when x=0, t=2; when x=2, t=0):I = integral from 2 to 0 of f(2 - t)(-dt) = integral from 0 to 2 of f(2 - t) dt.From the given functional equation f(x) = 1 - f(2 - x), we get f(2 - x) = 1 - f(x). Therefore:I = integral from 0 to 2 of (1 - f(t)) dt = integral from 0 to 2 of 1 dt - integral from 0 to 2 of f(t) dt = 2 - I.So I = 2 - I, giving 2I = 2, therefore I = 1.

Let f(x) be a continuous function from R to R such that f(x) = 1 - f(2 - x). Which one of the following options is the CORRECT value of the integral from 0 to 2 of f(x) dx?

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Let f(x) be a continuous function from R to R such that f(x) = 1 - f(2 - x). Which one of the following options is the CORRECT value of the integral from 0 to 2 of f(x) dx?

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