1. **State the problem:** We are given a function $f : \mathbb{R}^2 \to \mathbb{R}$ defined by the conditions: $$f(0,y) = y + 1$$ $$f(x+1,y) = f(x, f(x,y)) + x$$ We need to find the value of $f(2,2)$. 2. **Understand the function definition:** - The function is defined recursively. - The base case is $f(0,y) = y + 1$. - For $x+1$, the function depends on $f(x,y)$ and $f(x, f(x,y))$. 3. **Calculate $f(1,y)$:** Using the recursive formula: $$f(1,y) = f(0, f(0,y)) + 0 = f(0, y+1) = (y+1) + 1 = y + 2$$ 4. **Calculate $f(2,y)$:** $$f(2,y) = f(1, f(1,y)) + 1$$ From step 3, $f(1,y) = y + 2$, so: $$f(2,y) = f(1, y+2) + 1 = (y+2) + 2 + 1 = y + 5$$ 5. **Find $f(2,2)$:** $$f(2,2) = 2 + 5 = 7$$ **Final answer:** $$\boxed{7}$$