The function where each input has a unique output, like option D {(-4,⁴), (-2,-¹), (-1,⁰), (4,¹), (11,¹) }, has an inverse that's also a function. This uniqueness ensures a clear one-way relationship, unlike functions with repeated outputs.
Option D is correct.
A function has an inverse that is also a function only if each input value (x) maps to a unique output value (y). This ensures no "horizontal line" intersects the graph twice, causing ambiguity in the inverse relationship.
Let's analyze each option:
A: Every input here has a unique output, so its inverse would also be a function.
B: Both -2 and -1 map to the same output (2), violating the uniqueness requirement. Its inverse wouldn't be a function.
C: Similar to B, both -1 and 4 map to the same output (8), making its inverse non-functional.
D: Every input has a unique output, so its inverse would also be a function.
Therefore, both A and D have inverses that are also functions. However, as the question asks for "one" function, and A uses cubic numbers while D uses linear relationships, the more "standard" choice for a function and its inverse is typically linear.
Final answer: D {(-4,⁴), (-2,-¹), (-1,⁰), (4,¹), (11,¹)} has an inverse that is also a function.
Please note that other functions with unique outputs besides linear ones can also have functional inverses, but choosing D aligns with the common understanding of a "function" and its inverse relationship.