To find the domain of the function y = √(x + 6) - 7, which is the inverse of the equation y = 100 - x^2, we need to consider the restrictions on the domain of the original function and its inverse.
1. Original function: y = 100 - x^2
- The domain of the original function y = 100 - x^2 includes all real numbers because there are no restrictions on the input values of x.
2. Finding the inverse function:
- To find the inverse of y = 100 - x^2, we switch the roles of x and y and solve for y:
x = 100 - y^2
y^2 = 100 - x
y = ±√(100 - x)
Since the square root is usually taken as the positive value, the inverse function is y = √(100 - x).
3. New function: y = √(x + 6) - 7
- Comparing this with the inverse function y = √(100 - x), we see that x + 6 is equivalent to 100 - x.
- Solving x + 6 = 100 - x gives x = 47.
- Therefore, the domain of the function y = √(x + 6) - 7 is all real numbers where x is greater than or equal to 47, as x cannot be less than 47 due to the square root operation in the function.
In summary, the domain of the function y = √(x + 6) - 7, which is the inverse of y = 100 - x^2, is x ≥ 47.
