To determine the inverse of the function [tex]\( f(x) = \frac{1}{3}x - 2 \)[/tex], we need to follow these steps:
1. Express [tex]\( f(x) \)[/tex] as [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1}{3}x - 2
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse:
[tex]\[
x = \frac{1}{3}y - 2
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
- First, add 2 to both sides:
[tex]\[
x + 2 = \frac{1}{3}y
\][/tex]
- Then, multiply both sides by 3 to isolate [tex]\( y \)[/tex]:
[tex]\[
3(x + 2) = y
\][/tex]
4. Simplify the equation:
[tex]\[
y = 3(x + 2)
\][/tex]
This gives us the inverse function:
[tex]\[
f^{-1}(x) = 3x + 6
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{A. \, f^{-1}(x) = 3x + 6}
\][/tex]
