Question

For h(x, y)=sin^(-1)(x^2+y^2-7) the domain of the function is the area between two circles. Show your answers to 4 decimals if necessary. The larger circle has a radius of The s

Answer 1

The domain of the function h(x, y) is determined by finding the set of (x, y) that satisfy -1 ≤ x^2 + y^2 - 7 ≤ 1. This leads to the area between two circles with radii √6 and √8, approximately 2.4495 and 2.8284 respectively.

Step-by-step explanation:

The question asks us to determine the domain of the function h(x, y) = sin-1(x2 + y2 - 7). The domain of the inverse sine function, sin-1(x), is the set of all values that x can have for which the output is a real number, which is between -1 and 1. Therefore, for h(x, y) to be real-valued, we need x2 + y2 - 7 to be between -1 and 1.

This inequality can be expressed as:

• -1 ≤ x2 + y2 - 7 ≤ 1

These inequalities describe the area between two circles centered at the origin (0,0) on the xy-plane. The first inequality (-1 ≤ x2 + y2 - 7) can be rearranged to x2 + y2 ≤ 8, which defines a circle with a radius of √8. The second inequality (x2 + y2 - 7 ≤ 1) rearranges to x2 + y2 ≥ 6, defining a circle with a radius of √6. The domain is, therefore, the set of all points (x, y) that lie between these two circles with radii √6 and √8.

The radii can be approximated to four decimal places: radii 2.4495 (for √6) and 2.8284 (for √8).