To find the inverse of the function [tex]\( f(x)=\frac{3x}{5}+3 \)[/tex], we need to follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = \frac{3x}{5} + 3 \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function. The goal is to solve for [tex]\( y \)[/tex]:
[tex]\[ x = \frac{3y}{5} + 3 \][/tex]
3. Isolate [tex]\( y \)[/tex]. Start by subtracting 3 from both sides to isolate the term containing [tex]\( y \)[/tex]:
[tex]\[ x - 3 = \frac{3y}{5} \][/tex]
4. Multiply both sides by 5 to eliminate the fraction:
[tex]\[ 5(x - 3) = 3y \][/tex]
5. Solve for [tex]\( y \)[/tex] by dividing both sides by 3:
[tex]\[ y = \frac{5(x - 3)}{3} \][/tex]
Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{5(x - 3)}{3} \][/tex]
Therefore, the correct option is:
C. [tex]\( f^{-1}(x) = \frac{5(x - 3)}{3} \)[/tex]
