Final answer:
The domain and range of the equation representing a circle with center (4, 1) and radius 5 are the intervals [-1, 9] and [-4, 6], respectively.
Step-by-step explanation:
The student has asked about finding the domain and range of the given equation (x-4)²+(y-1)²= 25. This equation represents a circle in the Cartesian coordinate system with the center at point (4, 1) and a radius of 5.
The domain of the circle is the set of all x-values that the circle occupies on the x-axis. For this circle, the domain includes all x-values from (4 - 5) to (4 + 5), which are [-1, 9]. The range of the circle is the set of all y-values that the circle occupies on the y-axis. Similarly, for this circle, the range includes all y-values from (1 - 5) to (1 + 5), which are [-4, 6].
Hence, the domain of the relation is from -1 to 9, and the range is from -4 to 6.