To identify the inverse [tex]\( g(x) \)[/tex] of the given relation [tex]\( f(x) \)[/tex], we must understand that the inverse of a relation essentially swaps each [tex]\( (x, y) \)[/tex] pair to [tex]\( (y, x) \)[/tex]. Let's go through each pair in the given [tex]\( f(x) \)[/tex] and find the corresponding pairs for [tex]\( g(x) \)[/tex]:
1. Starting with [tex]\( f(x) \)[/tex]:
- The first pair is [tex]\( (8, 3) \)[/tex]. To find the inverse, we swap the components to get [tex]\( (3, 8) \)[/tex].
- The second pair is [tex]\( (4, 1) \)[/tex]. Swapping the components gives [tex]\( (1, 4) \)[/tex].
- The third pair is [tex]\( (0, -1) \)[/tex]. Swapping the components gives [tex]\( (-1, 0) \)[/tex].
- The fourth pair is [tex]\( (-4, -3) \)[/tex]. Swapping the components gives [tex]\( (-3, -4) \)[/tex].
2. Now we list these swapped pairs to form [tex]\( g(x) \)[/tex]:
- [tex]\( g(x) = \{(3, 8), (1, 4), (-1, 0), (-3, -4)\} \)[/tex].
Therefore, after identifying and swapping each pair from [tex]\( f(x) \)[/tex], the correct inverse relation [tex]\( g(x) \)[/tex] is:
[tex]\[ g(x) = \{(3, 8), (1, 4), (-1, 0), (-3, -4)\} \][/tex]
So the correct choice for the inverse [tex]\( g(x) \)[/tex] is:
[tex]\[ \boxed{g(x) = \{(3, 8), (1, 4), (-1, 0), (-3, -4)\}} \][/tex]
