Question

The function f(x)=|x-1|+2 is shown on the graph. Use the drop-down menus to complete the statements. In order for f⁻¹(x) to be a function, the domain of f(x) can be restricted to Under this restriction, the domain of f⁻¹(x) would be having the same minimum value as the

Answer 1

To ensure that the inverse of the function f(x)=|x-1|+2 is also a function, we must restrict the domain of f(x) to either x ≥ 1 or x ≤ 1. The domain of f⁻¹(x) will then correspond to the range of the restricted f(x), starting from the minimum or maximum value accordingly.

Step-by-step explanation:

The question involves understanding and manipulating the function f(x)=|x-1|+2 and its inverse to ensure the inverse is also a function. In order for f⁻¹(x) to be a function, the domain of f(x) must be restricted in such a way that the vertical line test is satisfied for the inverse as well. Specifically, because the graph of f(x) is a V-shaped graph due to the absolute value, we need to choose either the increasing or the decreasing part of the graph. Therefore, to make f(x) one-to-one and its inverse a function, we restrict the domain to either x ≥ 1 or x ≤ 1, depending on which part of the V we decide to use.

If we restrict the domain to x ≥ 1, then f⁻¹(x) would have a domain corresponding to the range of the restricted f(x), which starts from the minimum value of f(x) on the restricted domain. Alternatively, if we restrict the domain to x ≤ 1, then f⁻¹(x) would have a domain from the maximum value of the restricted f(x) downwards.