To find the inverse of the function [tex]\( f(x) = \frac{1}{3}x + 2 \)[/tex], follow these steps:
1. Rewrite [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]:
[tex]\[
x = \frac{1}{3}y + 2
\][/tex]
Subtract 2 from both sides:
[tex]\[
x - 2 = \frac{1}{3}y
\][/tex]
To isolate [tex]\( y \)[/tex], multiply both sides of the equation by 3:
[tex]\[
3(x - 2) = y
\][/tex]
4. Simplify the expression:
[tex]\[
y = 3x - 6
\][/tex]
The inverse function [tex]\( h(x) \)[/tex] is therefore:
[tex]\[
h(x) = 3x - 6
\][/tex]
So the correct answer is:
[tex]\[ h(x) = 3x - 6 \][/tex]
