A function is invertible if it is

The correct option is bijective.

To understand why an invertible function must be bijective, let us break down the concepts of invertibility, injectivity, and surjectivity step-by-step.

1. What is an Invertible Function?
A function $f:A\to B$ mapping elements from set $A$ to set $B$ is said to be invertible if there exists a unique inverse function $f^{-1}:B\to A$ such that for every element in the domain and codomain, the application of both functions returns the original element. That is, $f^{-1}\left(f\right(x\left)\right)=x$ for all $x\in A$, and $f\left(f^{-1}\right(y\left)\right)=y$ for all $y\in B$.

For this inverse function $f^{-1}$ to be a well-defined function itself, it must satisfy two crucial requirements: every element in $B$ must map back to exactly one element in $A$.

2. Requirement of Injectivity (One-to-One)
A function is injective (or one-to-one) if distinct elements in the domain map to distinct elements in the codomain. If $f$ were not injective, two different elements $x_1$ and $x_2$ in $A$ would map to the same element $y$ in $B$. When we attempt to define the inverse, $f^{-1}\left(y\right)$ would have to map to both $x_1$ and $x_2$. This violates the fundamental definition of a function, which states that each input must have a single, unique output. Thus, $f$ must be injective.

3. Requirement of Surjectivity (Onto)
A function is surjective (or onto) if every element in the codomain $B$ is mapped to by at least one element in the domain $A$. If $f$ were not surjective, there would be some element $y$ in $B$ that has no pre-image in $A$. Consequently, when trying to define the inverse, there would be no target for $f^{-1}\left(y\right)$ in set $A$. A valid function must be defined for all elements in its domain, so $f^{-1}$ would not be a well-defined function. Thus, $f$ must be surjective.

4. Conclusion
Since a function must be both injective (to ensure uniqueness of the reverse mapping) and surjective (to ensure completeness of the reverse mapping) to have a well-defined inverse, it must be bijective (which is defined as being both injective and surjective).