Given the function [tex]\( g \)[/tex], we are provided with the following values for [tex]\( x \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & -8 & -4 & 0 & 4 & 8 \\
\hline
g(x) & 2 & 3 & 4 & 5 & 6 \\
\hline
\end{array}
\][/tex]
We need to determine which of the following given functions could be the inverse of [tex]\( g \)[/tex].
---
To find the inverse of [tex]\( g \)[/tex], we need a function that reverses the outputs back into the original inputs. Mathematically, if [tex]\( g(a) = b \)[/tex], then [tex]\( g^{-1}(b) = a \)[/tex].
We will check each option to identify which one satisfies this condition:
### Option A:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 2 & 3 & 4 & 5 & 6 \\
\hline
k(x) & -8 & -4 & 0 & 4 & 8 \\
\hline
\end{array}
\][/tex]
For [tex]\( k \)[/tex] to be the inverse of [tex]\( g \)[/tex]:
- [tex]\( g(-8) = 2 \)[/tex] implies [tex]\( k(2) = -8 \)[/tex]
- [tex]\( g(-4) = 3 \)[/tex] implies [tex]\( k(3) = -4 \)[/tex]
- [tex]\( g(0) = 4 \)[/tex] implies [tex]\( k(4) = 0 \)[/tex]
- [tex]\( g(4) = 5 \)[/tex] implies [tex]\( k(5) = 4 \)[/tex]
- [tex]\( g(8) = 6 \)[/tex] implies [tex]\( k(6) = 8 \)[/tex]
Clearly, [tex]\( k(x) \)[/tex] matches these requirements exactly. Therefore, option A is a valid choice.
### Checking Other Options for Completeness
#### Option B:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & -8 & -4 & 0 & 4 & 8 \\
\hline
j(x) & -2 & -3 & -4 & 5 & 6 \\
\hline
\end{array}
\][/tex]
- [tex]\( g(-8) = 2 \)[/tex] so we need [tex]\( j(2) = -8 \)[/tex]
- [tex]\( g(-4) = 3 \)[/tex] so we need [tex]\( j(3) = -4 \)[/tex]
- [tex]\( g(0) = 4 \)[/tex] so we need [tex]\( j(4) = 0 \)[/tex]
- [tex]\( g(4) = 5 \)[/tex] so we need [tex]\( j(5) = 4 \)[/tex]
- [tex]\( g(8) = 6 \)[/tex] so we need [tex]\( j(6) = 8 \)[/tex]
These requirements are not matched by [tex]\( j(x) \)[/tex], so option B is not valid.
#### Option C:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 2 & 3 & 0 & 4 & 8 \\
\hline
m(x) & 8 & 4 & -4 & -5 & -6 \\
\hline
\end{array}
\][/tex]
The [tex]\( x \)[/tex] values don't match the outputs of [tex]\( g(x) \)[/tex] as inputs adequately, so [tex]\( m \)[/tex] cannot be the inverse.
#### Option D:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & -8 & -4 & 0 & 4 & 8 \\
\hline
h(x) & -2 & -3 & -4 & -5 & -6 \\
\hline
\end{array}
\][/tex]
Here, [tex]\( h(x) \)[/tex] does not match the patterns needed to reverse [tex]\( g(x) \)[/tex], so option D is not valid.
---
Thus, the function that could be the inverse of [tex]\( g \)[/tex] is:
[tex]\[
\boxed{\text{A}}
\][/tex]
