To determine the inverse of the function \( f(x) = \frac{3x}{5} + 3 \), let's go through the process of finding the inverse step-by-step:
1. Start with the given function:
[tex]\[
f(x) = \frac{3x}{5} + 3
\][/tex]
2. Replace \( f(x) \) with \( y \):
[tex]\[
y = \frac{3x}{5} + 3
\][/tex]
3. Swap \( x \) and \( y \):
[tex]\[
x = \frac{3y}{5} + 3
\][/tex]
4. Solve for \( y \):
- Subtract 3 from both sides:
[tex]\[
x - 3 = \frac{3y}{5}
\][/tex]
- Multiply both sides by 5 to eliminate the fraction:
[tex]\[
5(x - 3) = 3y
\][/tex]
- Divide both sides by 3 to isolate \( y \):
[tex]\[
y = \frac{5(x - 3)}{3}
\][/tex]
Therefore, the inverse function is:
[tex]\[
f^{-1}(x) = \frac{5(x - 3)}{3}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{A. \, f^{-1}(x) = \frac{5(x-3)}{3}}
\][/tex]
