What can be concluded if a function f has both a left inverse and a right inverse?

Steps to Solve

1. Define Left Inverse

A function $f: A \to B$ has a left inverse $g: B \to A$ if for every element $y \in B$, it holds that $g(f(x)) = x$ for some $x \in A$. This means that applying $g$ after $f$ retrieves the original input $x$.

2. Define Right Inverse

A function $f: A \to B$ has a right inverse $h: B \to A$ if for every element $x \in A$, it holds that $f(h(y)) = y$ for some $y \in B$. This indicates that applying $f$ after $h$ retrieves the original output $y$.

3. Connection to Injectivity and Surjectivity

If a function has a left inverse, then it is injective (one-to-one). This is derived from the fact that if $f(x_1) = f(x_2)$, applying $g$ gives $x_1 = x_2$.

If a function has a right inverse, then it is surjective (onto), as every element in $B$ has a pre-image in $A$.

4. Combining Inverses to Determine Bijectivity

If a function has both a left inverse and a right inverse, then it is both injective and surjective. Therefore, it is bijective, meaning every output in $B$ corresponds to exactly one input in $A$.

5. Conclusion about the Function

We conclude that if a function $f$ has both a left inverse and a right inverse, it must be a bijection. This means it is both injective and surjective.