Final answer:
The cosine function is not one-to-one across its domain and not a bijection. The greatest integer function is also not one-to-one and not a bijection. The piecewise function is not one-to-one due to the discontinuity at x = 1, and therefore not a bijection.
Step-by-step explanation:
To determine if the given functions are one-to-one and map onto their indicated codomains and thus check if they are bijections, we apply the definitions and perform the necessary tests for each function.
Function a: f(x) = cos(x)
The cosine function is not one-to-one on the entire real line because it repeats values in each period of 2π. For example, cos(0) = cos(2π) = 1. Since it fails the horizontal line test, it cannot be a bijection. Although it maps to the interval (-1, 1), it does not cover every value exactly once.
Function b: Greatest Integer Function int : R ⇒ Z
The greatest integer function, also known as the floor function, is not one-to-one because multiple real numbers can map to the same integer. For instance, int(1.5) = int(1.9) = 1. However, it maps onto Z because every integer has a corresponding real-number pre-image. It is also not a bijection.
Function c: Piecewise Function
For the piecewise function, we consider each segment individually:
2 - x for x < 1: This segment is a line with a negative slope and is one-to-one.
1/x for x > 1: This segment is a hyperbola and is also one-to-one.
However, combining these two segments does not provide a one-to-one mapping across the entire domain of real numbers due to the discontinuity at x = 1, hence not a bijection.