To find the inverse of the function [tex]\( f(x) = \frac{1}{3}x + 2 \)[/tex], follow these detailed steps:
1. Start with the function [tex]\( f(x) \)[/tex]:
[tex]\[
y = \frac{1}{3}x + 2
\][/tex]
Here, [tex]\( y \)[/tex] represents [tex]\( f(x) \)[/tex].
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to begin finding the inverse:
[tex]\[
x = \frac{1}{3}y + 2
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
- First, isolate the term containing [tex]\( y \)[/tex] by subtracting 2 from both sides:
[tex]\[
x - 2 = \frac{1}{3}y
\][/tex]
- Next, eliminate the fraction by multiplying both sides by 3:
[tex]\[
3(x - 2) = y
\][/tex]
4. Rewrite the equation to express [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex]:
[tex]\[
y = 3(x - 2)
\][/tex]
5. Simplify the right-hand side:
[tex]\[
y = 3x - 6
\][/tex]
So, the inverse function [tex]\( h(x) \)[/tex], which we denote by [tex]\( f^{-1}(x) \)[/tex], is:
[tex]\[
h(x) = 3x - 6
\][/tex]
Therefore, the correct answer is:
[tex]\[
h(x) = 3x - 6
\][/tex]
