Answer:
[0, ∞)
Step-by-step explanation:
The domain of the inverse function f⁻¹(x) is the range of the original function f(x), and the range of the inverse function f⁻¹(x) is the domain of the original function f(x). Therefore, to find the domain of y = f⁻¹(x) if f(x) = 1/2(x - 1)², we need to find the range of f(x).
The function f(x) is a quadratic function because its highest power of x is 2, which means it forms a parabola. Since the leading coefficient of f(x) is positive, the parabola opens upwards and has a minimum value at its vertex.
The function f(x) is already in vertex form, f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. Here, h = 1 and k = 0, so the vertex is (1, 0). This means that the function f(x) can take on any value greater than or equal to zero. Therefore, the range of f(x) is [0, ∞).
Since the domain of the inverse function f⁻¹(x) is the range of the original function f(x), the domain of y = f⁻¹(x) is:
[tex]\LARGE\boxed{[0,\infty)}[/tex]
