Sure, let's find the inverse of the given function [tex]\( f(x) = \frac{1}{3} x + 2 \)[/tex].
1. Rewrite the function: Start by replacing [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1}{3} x + 2
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]: To find the inverse function, we switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = \frac{1}{3} y + 2
\][/tex]
3. Solve for [tex]\( y \)[/tex]: Isolate [tex]\( y \)[/tex] in the equation.
- First, subtract 2 from both sides to start isolating [tex]\( y \)[/tex]:
[tex]\[
x - 2 = \frac{1}{3} y
\][/tex]
- Next, multiply both sides of the equation by 3 to solve for [tex]\( y \)[/tex]:
[tex]\[
3(x - 2) = y
\][/tex]
- Simplify the right hand side:
[tex]\[
y = 3x - 6
\][/tex]
4. Write the inverse function: The function obtained is the inverse of [tex]\( f(x) \)[/tex]. We denote the inverse of [tex]\( f \)[/tex] as [tex]\( h(x) \)[/tex], therefore:
[tex]\[
h(x) = 3x - 6
\][/tex]
So, the inverse of the given function [tex]\( f(x) = \frac{1}{3} x + 2 \)[/tex] is [tex]\( \boxed{h(x) = 3x - 6} \)[/tex].
