Final answer:
The quadratic function f(x) = x² - x is not one-to-one unless its domain is restricted. The least integer value of k for which f is one-to-one, after examining its derivative and critical points, is k = 1.
Step-by-step explanation:
To determine if the function f(x) = x² - x is one-to-one, we need to check if for every value of y, there is only one corresponding x value. It is a well-known fact that a quadratic function is not one-to-one unless its domain is restricted. In this case, we have to find a value of k such that for all x ≥ k, the function is one-to-one.
To do this, we examine the derivative of f(x) to find its critical points, which will indicate where the function stops decreasing and starts increasing, or vice versa. The derivative of f(x) is f'(x) = 2x - 1. Setting this equal to zero gives us a critical point at x = 1/2. However, since we are looking for an integer value of k where the domain starts, we check x = 0 and x = 1. The function decreases on the interval (-∞, 1/2) and increases on the interval (1/2, ∞). Thus, the function will be one-to-one on either interval. Since we want the smallest integer value of k, we select k = 1.
Therefore, the correct answer to whether f is one-to-one for x ≥ k and the least value of k is b) No; k = 1.