To find the inverse of the function [tex]\( f(x) = \frac{2x - 5}{3} \)[/tex], follow these detailed steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = \frac{2x - 5}{3} \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = \frac{2y - 5}{3} \][/tex]
3. Solve for [tex]\( y \)[/tex]:
a. Multiply both sides by 3 to eliminate the denominator:
[tex]\[ 3x = 2y - 5 \][/tex]
b. Add 5 to both sides to isolate the term with [tex]\( y \)[/tex]:
[tex]\[ 3x + 5 = 2y \][/tex]
c. Divide both sides by 2 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{3x + 5}{2} \][/tex]
4. Therefore, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{3x + 5}{2} \][/tex]
Among the given options, the correct answer is:
[tex]\[ \boxed{\frac{3x + 5}{2}} \][/tex]
