Final answer:
The question involves finding the inverse of a given function by restricting its domain to ensure it is one-to-one. A horizontal line test is used to verify that the graph represents a one-to-one function. The domain should be restricted appropriately before the inverse can be found by swapping x and y and solving for y.
Step-by-step explanation:
The student's question pertains to the inverses of functions and how to restrict the domain of a function so that its inverse is also a function. When we have the function f(x), we need to ensure that it is a one-to-one function by restricting its domain so that every x-value has only one unique y-value. A horizontal line test can be applied to determine if a curve represents a one-to-one function, and if the line intersects the graph more than once, the domain should be restricted accordingly.
To find an inverse, we swap the x and y values and solve for y. For example, if f(x)=x3 - 4, to find f-1(x), we would set y = x3 - 4 and then solve for x:
x = y3 - 4 which could be rewritten as f-1(y) = y3 + 4 after switching x and y again. In this hypothetical case, the domain should be restricted such that the original function passes the horizontal line test, so it may be x ≥ 4 or x ≤ -4 depending on the graph's behavior.