31. Let f, g : R→R be defined, respectively by f(x) = x + 1, g(x) = 2x – 3. Find f + g, f – g and $\frac{f}{}$  .

Circle S? : x² + y² - 10x - 10y + 41 = 0.
Center C? = (5, 5). Radius r? = √ (5² + 5² - 41) = √ (25 + 25 - 41) = √9 = 3.
Circle S? : x² + y² - 16x - 10y + 80 = 0.
Center C? = (8, 5). Radius r? = √ (8² + 5² - 80) = √ (64 + 25 - 80) = √9 = 3.
The solution checks if the center of one circle lies on the other.
Put C? (8, 5) into S? : 8² + 5² - 10 (8) - 10 (5) + 41 = 64 + 25 - 80 - 50 + 41 = 130 - 130 = 0. So C? lies on S?
Put C? (5, 5) into S? : 5² + 5² - 16 (5) - 10 (5) + 80 = 25 + 25 - 80 - 50 + 80 = 130 - 130 = 0. So C? lies on S?
This means both circles pass through the center of each other. So statement (D) is correct.
The incorrect statement is asked.
Let's check other options.
(A) Distance between centers C? = √ (8-5)² + (5-5)²) = √3² = 3. Average of radii = (3+3)/2 = 3. Statement (A) is correct.
(C) Since C? = 3 and r? + r? = 6, and |r? - r? | = 0, we have |r? - r? | < C? < r? + r? This means the circles intersect at two points. Statement (C) is correct.
(B) Both circles' centres lie inside region of one another. C? is inside S? if the distance C? < r? Here, 3 = r? So C? is on S? , not inside. C? is on S? , not inside. So statement (B) is incorrect.