To solve this problem, we need to find the inverse [tex]\( g(x) \)[/tex] of the given relation [tex]\( f(x) \)[/tex] and determine whether [tex]\( g(x) \)[/tex] is a function.
1. Finding the inverse relation [tex]\( g(x) \)[/tex]:
Given the relation [tex]\( f(x) = \{(8, 3), (4, 1), (0, -1), (-4, -3)\} \)[/tex], to find the inverse, we swap the elements in each ordered pair.
Swapping each pair:
- From [tex]\( (8, 3) \)[/tex] to [tex]\( (3, 8) \)[/tex]
- From [tex]\( (4, 1) \)[/tex] to [tex]\( (1, 4) \)[/tex]
- From [tex]\( (0, -1) \)[/tex] to [tex]\( (-1, 0) \)[/tex]
- From [tex]\( (-4, -3) \)[/tex] to [tex]\( (-3, -4) \)[/tex]
Therefore, the inverse relation [tex]\( g(x) \)[/tex] is:
[tex]\[
g(x) = \{(3, 8), (1, 4), (-1, 0), (-3, -4)\}
\][/tex]
2. Choosing a true statement:
To determine whether [tex]\( g(x) \)[/tex] is a function, we must check if [tex]\( f(x) \)[/tex] is one-to-one.
- A relation is one-to-one if each element in the domain is paired with a unique element in the range.
- For [tex]\( f(x) \)[/tex], the pairs [tex]\((8, 3)\)[/tex], [tex]\((4, 1)\)[/tex], [tex]\((0, -1)\)[/tex], and [tex]\((-4, -3)\)[/tex] all have unique second elements (3, 1, -1, -3).
Since no two different elements in the domain of [tex]\( f(x) \)[/tex] map to the same element in the range, [tex]\( f(x) \)[/tex] is one-to-one.
Therefore, the inverse [tex]\( g(x) \)[/tex] must be a function because an inverse relation is a function if and only if the original relation is one-to-one.
The correct true statement is:
[tex]\[
\boxed{g(x) \text{ is a function because } f(x) \text{ is one-to-one.}}
\][/tex]
