State with reason whether the following function has an inverse: $g : \{5, 6, 7, 8\} \rightarrow \{1, 2, 3, 4\}$ with $g = \{(5, 4), (6, 3), (7, 4), (8, 2)\}$.
(NO) function $g$ has an inverse if and only if it is a bijection (both one-one and onto).
Given $g = \{(5, 4), (6, 3), (7, 4), (8, 2)\}$.
We observe that $g(5) = 4$ and $g(7) = 4$.
Since two distinct elements in the domain,$5$ and $7$,map to the same element $4$ in the codomain,the function $g$ is not one-one (it is many-one).
Because $g$ is not one-one,it is not a bijection.
Therefore,the function $g$ does not have an inverse.
Given $g = \{(5, 4), (6, 3), (7, 4), (8, 2)\}$.
We observe that $g(5) = 4$ and $g(7) = 4$.
Since two distinct elements in the domain,$5$ and $7$,map to the same element $4$ in the codomain,the function $g$ is not one-one (it is many-one).
Because $g$ is not one-one,it is not a bijection.
Therefore,the function $g$ does not have an inverse.
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