The graph of a quadratic function with vertex $(-4,1)$ is shown in the figure below. Find the range and the domain.
Write your answers as inequalities, using $x$ or $y$ as appropriate. Or, you may instead click on "Empty set" or "All reals" as the answer.

Step 1 ：The domain of a function is the set of all possible input values (often the 'x' variable), which produce a valid output from a particular function. For a quadratic function, the domain is all real numbers because a quadratic function is defined for all real numbers.

Step 2 ：The range of a function is the complete set of possible output values (usually the 'y' variable), which result from using the function formula. The range of a quadratic function depends on the direction of the parabola. If the parabola opens upwards, the range is $y\ge k$, where $k$ is the y-coordinate of the vertex. If the parabola opens downwards, the range is $y\le k$, where $k$ is the y-coordinate of the vertex.

Step 3 ：Since the vertex of the quadratic function is given as $(-4,1)$, and we don't have information about the direction of the parabola, we can't determine the range. We need more information about the function to determine the range.

Step 4 ：The domain of the function is all real numbers, which can be written as $-\mathrm{\infty }<x<\mathrm{\infty }$. The range of the function cannot be determined with the given information.

Step 5 ：$\boxed{{\displaystyle \text{The domain of the function is all real numbers, which can be written as }-\mathrm{\infty }<x<\mathrm{\infty }.\text{ The range of the function cannot be determined with the given information.}}}$