To find the inverse of the function [tex]\( f(x) = \frac{1}{3}x + 2 \)[/tex], we need to follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = \frac{1}{3}x + 2 \][/tex]
2. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
To find the inverse, we swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = \frac{1}{3}y + 2 \][/tex]
3. Solve for [tex]\( y \)[/tex]:
We need to isolate [tex]\( y \)[/tex] on one side of the equation.
- First, subtract 2 from both sides:
[tex]\[ x - 2 = \frac{1}{3}y \][/tex]
- Next, multiply both sides by 3 to solve for [tex]\( y \)[/tex]:
[tex]\[ 3(x - 2) = y \][/tex]
- Simplify:
[tex]\[ y = 3(x - 2) \][/tex]
4. Rewrite the inverse function:
[tex]\( y = 3x - 6 \)[/tex] is the inverse function. Therefore, we denote the inverse function as:
[tex]\[ f^{-1}(x) = 3x - 6 \][/tex]
So, the inverse of the function [tex]\( f(x) = \frac{1}{3}x + 2 \)[/tex] is:
[tex]\[ h(x) = 3x - 6 \][/tex]
Thus, the correct answer is:
[tex]\[ h(x) = 3x - 6 \][/tex]
