Final answer:
The inverse of the function y=x²+7, with the domain x ≥ 0, is y = √(x - 7), with the domain x ≥ 7 due to the constraints on taking square roots of non-negative numbers.
Step-by-step explanation:
The inverse of a function essentially switches the 'x' and 'y' in the original function, and then we solve for the new 'y'. In the case of y=x²+7, with the domain x ≥ 0, we first write the inverse relationship by swapping 'x' and 'y' to get x = y² + 7. Next, we solve for 'y' to find the inverse function:
- Subtract 7 from both sides: x - 7 = y².
- Since the domain of the original function restricts x to non-negative values (x ≥ 0), we only consider the positive square root: y = √(x - 7).
The domain restriction is crucial because without it the square root function would be multi-valued. Therefore, the inverse function is y = √(x - 7), with the domain x ≥ 7, because you can’t take the square root of a negative number (within the set of real numbers).