The given statement " , the function has a well-defined inverse.fr→r f(x)=2x" is true because the function f(x) = 2x is bijective and, therefore, possesses a well-defined inverse function f⁻¹(x), where f⁻¹(2x) = x.
True. For the function f(x) = 2x, let's examine whether it has a well-defined inverse. The function f takes a real number x and maps it to 2x. To ascertain if f has an inverse, we must check if f is one-to-one or injective, indicating that distinct inputs map to distinct outputs.
For f(x) = 2x, if x₁ and x₂ are distinct, then f(x₁) = 2x₁ and f(x₂) = 2x₂. As x₁ and x₂ are distinct, f(x₁) and f(x₂) are also distinct, demonstrating injectivity.
Next, we need to confirm that f is surjective, meaning that every element in the codomain is the image of some element in the domain. In this case, since f(x) = 2x, any real number y in the codomain can be expressed as y = 2x for some x. Therefore, f is surjective.
Since f is both injective and surjective, it is bijective and has a well-defined inverse. The inverse function, denoted as f⁻¹(x), would be such that f⁻¹(2x) = x.