Question

g = {(0, 1)} is ( g^{-1} ) a function and why?
○ no, there are only a limited number of ordered pairs and can’t be determined.
○ no, we don’t know the original function and therefore can’t make the determination.
○ yes, each element in the domain has only one range value.
○ yes, each element in the range has only one domain value.

Explanation:

To determine if \( G^{-1} \) is a function, we first find \( G^{-1} \). For a relation \( G = \{(x,y)\} \), its inverse \( G^{-1} \) is \( \{(y,x)\} \). Here, \( G = \{(0,1)\} \), so \( G^{-1}=\{(1,0)\} \). A relation is a function if each element in the domain (first elements of ordered pairs) has exactly one corresponding element in the range (second elements). For \( G^{-1}=\{(1,0)\} \), the domain has only 1, and it maps to 0 (only one range value). But more precisely, for the inverse of a function (or to check if the inverse is a function), we use the horizontal line test for the original, but for a single ordered pair, the inverse will have a single ordered pair. The key is that in \( G^{-1} \), each element in the range of \( G \) (which is the domain of \( G^{-1} \)) has only one domain element of \( G \) (range of \( G^{-1} \)). So \( G^{-1} \) is a function because each element in its domain (1) has exactly one range element (0), or equivalently, each element in the range of \( G \) (which is the domain of \( G^{-1} \)) has only one domain element of \( G \) (so the inverse relation has each domain element mapping to one range element). The correct option is the one that states "Yes, each element in the range has only one domain value" (since the range of \( G \) is the domain of \( G^{-1} \), and each element in that range (1) has only one domain value from \( G \) (0), so \( G^{-1} \) is a function as each domain element of \( G^{-1} \) maps to one range element).

Answer:

D. Yes, each element in the range has only one domain value.