Question

Let \( f(x) \) be a continuous function from \( \mathbb{R} \) to \( \mathbb{R} \) 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

Hint:

Use symmetry properties of functions and substitution to simplify definite integrals.

Answer 1

The Correct Option is B

The given functional equation is: \[ f(x) + f(2 - x) = 1 \] Step 1: Split the integral.
The integral can be split as: \[ \int_0^2 f(x) \, dx = \int_0^1 f(x) \, dx + \int_1^2 f(x) \, dx \] Step 2: Use substitution for symmetry.
In the second term, substitute \( u = 2 - x \): \[ \int_1^2 f(x) \, dx = \int_0^1 f(2 - u) \, du \] Using \( f(2 - u) = 1 - f(u) \), this becomes: \[ \int_1^2 f(x) \, dx = \int_0^1 (1 - f(u)) \, du \] Step 3: Simplify the integral.
\[ \int_0^2 f(x) \, dx = \int_0^1 f(x) \, dx + \int_0^1 1 \, du - \int_0^1 f(u) \, du \] \[ \int_0^2 f(x) \, dx = \int_0^1 1 \, du = 1 \] Final Answer: \[ \boxed{1} \]