To find the inverse of the function [tex]\( f(x) = \frac{1}{3}x + 2 \)[/tex], we need to swap the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] and then solve for [tex]\( y \)[/tex].
1. Start with the equation:
[tex]\[
y = f(x) = \frac{1}{3}x + 2
\][/tex]
2. To find the inverse, interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = \frac{1}{3}y + 2
\][/tex]
3. Solve for [tex]\( y \)[/tex]. First, isolate [tex]\( y \)[/tex]:
[tex]\[
x - 2 = \frac{1}{3}y
\][/tex]
4. Multiply both sides by 3 to solve for [tex]\( y \)[/tex]:
[tex]\[
3(x - 2) = y
\][/tex]
5. Simplify the expression:
[tex]\[
y = 3x - 6
\][/tex]
So, the inverse function [tex]\( h(x) \)[/tex] is [tex]\( 3x - 6 \)[/tex].
Thus, the correct answer from the given options is:
[tex]\[
h(x) = 3x - 6
\][/tex]
