Final answer:
The task is to evaluate the bijection property of different functions from ℝ to ℝ. Linear and exponential functions are bijections, while quadratic functions are not. The inverse function can be a bijection with domain restrictions.The correct answer is option C.
Step-by-step explanation:
The question is asking to determine if each function provided is a bijection from ℝ to ℝ, where ℝ represents the set of all real numbers. To be a bijection, a function must be both injective (one-to-one) and surjective (onto). An injective function means that every element of the domain maps to a unique element in the codomain, without any two elements in the domain mapping to the same element in the codomain. A surjective function means that every element in the codomain is the image of at least one element in the domain.
A linear function, which is likely represented by a straight line with a non-zero slope, is bijective as each value of x maps to a unique value of y, and every possible y-value has a corresponding x-value. A quadratic function (typically a parabola) is not a bijection since it is not injective; it fails the horizontal line test as some y-values correspond to two different x-values. An inverse function, such as y = 1/x, is a bijection if its domain is restricted to exclude zero, as then each x-value will have a unique y-value and vice versa. Lastly, an exponential function (for example, y = e^x where e is the base of natural logarithms) is also a bijection since it is both injective and surjective on the set of real numbers.