To identify the inverse [tex]\( g(x) \)[/tex] of the given relation [tex]\( f(x) \)[/tex], we need to determine which of the provided options correctly represents [tex]\( g(x) \)[/tex].
Given relation:
[tex]\[ f(x) = \{(8, 3), (4, 1), (0, -1), (-4, -3)\} \][/tex]
The inverse of a relation is obtained by swapping the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values in each pair of the original relation. Let's swap the values for each pair in [tex]\( f(x) \)[/tex]:
1. Switching [tex]\( (8, 3) \)[/tex] gives [tex]\( (3, 8) \)[/tex]
2. Switching [tex]\( (4, 1) \)[/tex] gives [tex]\( (1, 4) \)[/tex]
3. Switching [tex]\( (0, -1) \)[/tex] gives [tex]\( (-1, 0) \)[/tex]
4. Switching [tex]\( (-4, -3) \)[/tex] gives [tex]\( (-3, -4) \)[/tex]
Therefore, the inverse relation [tex]\( g(x) \)[/tex] should be:
[tex]\[ g(x) = \{(3, 8), (1, 4), (-1, 0), (-3, -4)\} \][/tex]
Now let's compare this with the given options:
1. [tex]\( g(x) = \{(-4, -3), (0, -1), (4, 1), (8, 3)\} \)[/tex]
2. [tex]\( g(x) = \{(-8, -3), (-4, 1), (0, 1), (4, 3)\} \)[/tex]
3. [tex]\( g(x) = \{(8, -3), (4, -1), (0, 1), (-4, 3)\} \)[/tex]
4. [tex]\( g(x) = \{(3, 8), (1, 4), (-1, 0), (-3, -4)\} \)[/tex]
The correct inverse relation is [tex]\( g(x) = \{(3, 8), (1, 4), (-1, 0), (-3, -4)\} \)[/tex], which matches option 4.
Thus, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
