Linear Algebra: The inverse of an injective function

Question

I have this question from multiple choice section in an old exam paper in Linear Algebra:

As I read it, what they are saying is that if $f$ is injective then there exists an inverse function $g$.

This quote from the book Linear Algebra Done Right contradicts this:

since injective is a nessesary but not sufficient criterion for invertibility.

I am unsure if I am reading the question right though.

Answer 1

As I read it, what they are saying is that if $ f $ is injective then there exists an inverse function $ g $.

That is not quite true. Notice that the domain of $ g $ is not equal to the codomain of $ f $, so $ g $ is not an inverse function of $ f $. Instead, it is an inverse of $ f' : \mathbb{R}^3 \to \text{im} (f) $, which does not contradict the quote from Linear Algebra Done Right, since $ f' $ is injective (because $ f $ is already injective) and surjective (because we defined its codomain to be equal to its image).