Mathematics

This problem is about understanding when a function can be inverted by restricting its domain. The key insight is that a function only has an inverse if it's one-to-one (injective)—each output comes from exactly one input. By restricting $g$ to $x \in ]-\infty, -3]$, you're cutting off the parabola at its vertex, making it monotonic. Once you verify this restriction creates a one-to-one function, you can find the inverse algebraically by swapping $x$ and $y$, then solving for $y$. The domain and range of the inverse are simply the range and domain of the original function, swapped.

This problem is about understanding when a function can be inverted by restricting its domain. The key insight is that a function only has an inverse if it's one-to-one (injective)—each output comes from exactly one input. By restricting $g$ to $x \in ]-\infty, -3]$, you're cutting off the parabola at its vertex, making it monotonic. Once you verify this restriction creates a one-to-one function, you can find the inverse algebraically by swapping $x$ and $y$, then solving for $y$. The domain and range of the inverse are simply the range and domain of the original function, swapped.