For a function $f:A\to B$ to have an inverse function $f^{-1}:B\to A$, it must be a bijection. This means it must be both:
One-to-one (injective): Every element in $A$ maps to a unique element in $B$. (No two elements in $A$ share the same image).
Onto (surjective): Every element in $B$ is mapped to by at least one element in $A$. (The codomain equals the range).
A. If $f$ is one-to-one, then $f$ must be invertible.
FALSE. Being one-to-one is only half the requirement. If $f$ is not onto, there will be elements in $B$ that have no corresponding element in $A$. The inverse "function" would be undefined for these elements, so a true inverse function from $B$ to $A$ cannot exist.
B. If $f$ is onto, then $f$ must be invertible.
FALSE. Again, this is only half the requirement. If $f$ is onto but not one-to-one, multiple elements in $A$ map to the same element in $B$. The inverse relation would have one input mapping to multiple outputs, which violates the definition of a function.
C. If $f$ has an inverse function $f^{-1}:B\to A$, then every element of $B$ is the image of exactly one element of $A$.
TRUE. If $f$ has an inverse, it is a bijection. "Every element of $B$ is the image..." describes the onto property. "...of exactly one element of $A$" describes the one-to-one property. This is the exact definition of a bijective function.
D. If $f$ is not onto, then $f$ can still have an inverse function $f^{-1}:B\to A$.
FALSE. If $f$ is not onto, there is at least one element $y$ in $B$ that does not come from any $x$ in $A$. Therefore, $f^{-1}(y)$ would be undefined. Since a function $f^{-1}:B\to A$ must be defined for every element in its domain $B$, it cannot be a function.
