To find the inverse of the function [tex]\( f(x) = \frac{1}{3} x + 2 \)[/tex], we need to follow these steps:
1. Rewrite the function [tex]\( f(x) \)[/tex] as [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1}{3} x + 2
\][/tex]
2. Solve this equation for [tex]\( x \)[/tex]. To do this, we will isolate [tex]\( x \)[/tex] on one side of the equation.
3. Subtract 2 from both sides:
[tex]\[
y - 2 = \frac{1}{3} x
\][/tex]
4. Multiply both sides of the equation by 3 to eliminate the fraction:
[tex]\[
3(y - 2) = x
\][/tex]
5. Rewrite the expression by swapping [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to represent the inverse function [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[
f^{-1}(x) = 3(x - 2)
\][/tex]
6. Simplify the expression if necessary:
[tex]\[
f^{-1}(x) = 3x - 6
\][/tex]
Thus, the inverse function of [tex]\( f(x) = \frac{1}{3} x + 2 \)[/tex] is [tex]\( f^{-1}(x) = 3x - 6 \)[/tex].
Therefore, the correct answer is:
[tex]\[ h(x) = 3x - 6 \][/tex]
