Answer:
The inverse of a square root function is not strictly a quadratic function but rather half of the parabolic curve of the quadratic function.
Explanation:
The inverse of a square root function represents one half of the parabolic curve of a quadratic function. It is not strictly a quadratic function but rather a segment of the graph of its inverse quadratic function. This distinction arises due to the domain restrictions of the square root function.
The domain of a square root function is restricted to values of x that ensure the expression under the square root is non-negative. This restriction is because the square root of a negative number is undefined in the real number system. Consequently, the range of the square root function is limited to values of y that satisfy this constraint.
The inverse of a function is the reflection of the original function across the line y = x. The domain of the original function corresponds to the range of the inverse function, and vice versa. Therefore, as the domain and range of the square root function are restricted, so too are the domain and range of the quadratic function that is its inverse. By restricting the domain (and range) of the quadratic function in this manner, the two functions become precise reflections of each other across the line y = x.
In summary, the inverse of a square root function is a quadratic function whose domain is restricted to the range of the square root function.
Additional Notes
When determining the inverse of a function f, our objective is to identify another function that can reverse the mapping of f(x) back to x. However, in the case of a quadratic function, there exist two distinct values of x that produce the same y-value. To address this challenge, we typically constrain the domain of the quadratic function, often limiting it to values of x on one side of the axis of symmetry. This restricted domain allows us to create a modified version of f(x) that can have a well-defined inverse.