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 ...

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 int_0^2 f(x) dx?

A. 0
B. 1
C. 2
D. -1

Accepted Answer

1 (option B) The functional equation f(x) = 1 - f(2-x) means that f and its reflection about x = 1 sum to 1 at every point. When we integrate both sides over [0, 2] and use the substitution u = 2-x on the right-hand integral, we find that int_0^2 f(x) dx appears on both sides, allowing us to solve for it directly. This is a classic symmetry argument for definite integrals.

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 int_0^2 f(x) dx? A. 0 B. 1 C. 2 D. -1