Final answer:
The range of the inverse function f⁻¹(x) of the one-to-one function f(x) = (4x - 1) / (x + 2) is all real numbers except y ≠ -2.
Step-by-step explanation:
The given function f(x) = (4x - 1) / (x + 2) is a one-to-one function. To find the range of the inverse function f⁻¹(x), we need to determine the range of the original function f(x). Since f(x) is a rational function, its range will be all real numbers except for any value that makes the denominator zero. In this case, the denominator x + 2 becomes zero when x = -2. Therefore, the range of f(x) is all real numbers except x ≠ -2.
To find the range of the inverse function, we essentially find the domain of the original function since the domain and range switch places for a function and its inverse. Since the domain of f(x) was all real numbers except x ≠ -2, the range of f⁻¹(x) will be all real numbers except y ≠ -2 (since y = f(x) for the original function).