1. The problem asks to identify which of the given relations are functions. 2. A relation is a function if every input $x$ has exactly one output $y$. 3. Analyze each relation: - A: $f = \{(x, y) \mid x^2 + y^2 = 49, -10 \leq x \leq 0, y \geq 0\}$ - This is a semicircle of radius 7 centered at origin, restricted to $x$ from -10 to 0 and $y \geq 0$. - For some $x$ values, there is exactly one $y$ (since $y \geq 0$ restricts to upper semicircle). - So $f$ is a function. - B: $g = \{(x, y) \mid y = x^{2022}, x, y > 0\}$ - The function $y = x^{2022}$ is a function because for each $x > 0$, there is exactly one $y$. - C: $h = \{(x, y) \mid x = y^2, y > 0\}$ - Here $x$ is expressed in terms of $y$, but to check if $h$ is a function from $x$ to $y$, for each $x$ there must be one $y$. - Since $x = y^2$, for each $x > 0$, there is exactly one $y > 0$ (positive root), so $h$ is a function. - D: $k = \{(x, y) \mid y = 2x + (x - 1)^2, x > 0, y > 0\}$ - This is a polynomial expression for $y$ in terms of $x$. - For each $x > 0$, there is exactly one $y$. - So $k$ is a function. 4. Conclusion: All relations B, C, and D are functions. Relation A is also a function because the restriction $y \geq 0$ ensures one $y$ for each $x$. Final answer: All relations A, B, C, and D are functions.