Question:

To determine the correct statements, we analyze the given expressions for $f(x)$ and $g(x)$. Step 1: Evaluate $f(x)$. Consider the expression for $f(x)$ given in one of the options: $$f(x) = \frac12 \ln(x^2-2x+2)$$ This expression is consistent with the antiderivative of $\frac{x-1}{x^2-2x+2}$, as: $$\frac{d}{dx} \left( \frac12 \ln(x^2-2x+2) \right) = \frac12 \cdot \frac{2x-2}{x^2-2x+2} = \frac{x-1}{x^2-2x+2}$$ Thus, the statement $f(x) = \frac12 \ln(x^2-2x+2)$ is a correct expression for $f(x)$ under a suitable definition. Step 2: Evaluate $g(x)$. Consider the expression for $g(x)$ given in one of the options: $$g(x) = \frac12 \arctan(x-1)$$ This expression is consistent with the antiderivative of $\frac{1}{2(x^2-2x+2)}$, as: $$\frac{d}{dx} \left( \frac12 \arctan(x-1) \right) = \frac12 \cdot \frac{1}{1+(x-1)^2} = \frac{1}{2(x^2-2x+2)}$$ Thus, the statement $g(x) = \frac12 \arctan(x-1)$ is a correct expression for $g(x)$ under a suitable definition. Step 3: Evaluate $f(x)+g(x)$. Using the expressions for $f(x)$ and $g(x)$ established in Step 1 and Step 2: $$f(x)+g(x) = \frac12 \ln(x^2-2x+2) + \frac12 \arctan(x-1)$$ This sum matches the form of the expression for $f(x)+g(x)$ in one of the options, assuming a coefficient of $\frac12$ for the $\arctan(x-1)$ term. Based on these evaluations, the statements $f(x) = \frac12 \ln(x^2-2x+2)$, $g(x) = \frac12 \arctan(x-1)$, and their sum $f(x)+g(x) = \frac12 \ln(x^2-2x+2) + \frac12 \arctan(x-1)$ are consistent.