Final answer:
A bijective function is both injective and surjective, where each element in the domain maps to a unique element in the co-domain, and vice versa. To prove that a function is bijective and to find specific function values, the explicit form of the function is required. Without this information, it is impossible to proceed with the proof or to determine f(2) and the inverse f⁻¹(1).
Step-by-step explanation:
Defining a Bijective Function
A bijective function is a type of function that is both injective (one-to-one) and surjective (onto). In simple terms, every element in the domain of the function corresponds to a unique element in the co-domain, and every element in the co-domain is an image of some element in the domain. To prove that a function f(x) is bijective, you must demonstrate that it is both injective and surjective.
However, in order to prove that a function f is bijective, and find f(2) and f⁻¹(1), we need the specific form of the function f(x), which is missing in the provided question. Since this detail is crucial for the proof and finding the values, without it we cannot continue with the proof or calculations.
The provided information about a function f(x) could indicate that the graph of f(x) is a horizontal line, suggesting that f(x) takes on a constant value. A constant function can be bijective if its domain and co-domain contain precisely one element. However, in a more general case with a larger domain and co-domain, a constant function cannot be bijective as it would not be injective.
Without the specific form of f(x), we also cannot find the value of f(2) nor can we find the inverse function f⁻¹(1). If f(x) were bijective, the inverse function f⁻¹ would exist such that f(f⁻¹(y)) = y for every y in the co-domain of f.