Final answer:
The function with a well-defined inverse from the given options is f(x) = e⁻¹, as it is both injective and surjective, which are required properties for a function to have a well-defined inverse.
Step-by-step explanation:
The function among the given options that has a well-defined inverse is f(x) = e⁻¹. To have a well-defined inverse, a function must be both injective (one-to-one) and surjective (onto), meaning every element in the function's codomain is mapped to by exactly one element in its domain. The function f(x) = e⁻¹ satisfies this condition because it is a strictly increasing function, which guarantees it to be one-to-one, and it maps the set of all real numbers onto the set of all positive real numbers, making it onto.
For comparison:
- Option 1, f(x) = x² is not one-to-one over the real numbers because both positive and negative values of 'x' will yield the same 'x²'.
- Option 2, f(x) = |x| is not one-to-one as well because it produces the same value for both positive and negative inputs.
- Option 3, f(x) = sin(x) is also not one-to-one because the sine function repeats every 2π radians, producing the same output for multiple inputs.
Knowing these properties can help students understand more about functions, their inverses, and how they can be useful in different mathematical problems, such as solving for a variable in an equation.