Answer :
To determine the inverse of the function \( f(x) = \frac{2x - 3}{5} \), we need to follow these steps:
1. Replace \( f(x) \) with \( y \):
[tex]\[ y = \frac{2x - 3}{5} \][/tex]
2. Interchange \( x \) and \( y \):
[tex]\[ x = \frac{2y - 3}{5} \][/tex]
3. Solve this equation for \( y \):
[tex]\[ x = \frac{2y - 3}{5} \][/tex]
To solve for \( y \), we first eliminate the fraction by multiplying both sides by 5:
[tex]\[ 5x = 2y - 3 \][/tex]
4. Isolate \( y \):
[tex]\[ 5x + 3 = 2y \][/tex]
[tex]\[ y = \frac{5x + 3}{2} \][/tex]
Therefore, the inverse function is:
[tex]\[ f^{-1}(x) = \frac{5x + 3}{2} \][/tex]
Based on the given choices, the correct answer is:
D. [tex]\( f^{-1}(x) = \frac{5x + 3}{2} \)[/tex]
1. Replace \( f(x) \) with \( y \):
[tex]\[ y = \frac{2x - 3}{5} \][/tex]
2. Interchange \( x \) and \( y \):
[tex]\[ x = \frac{2y - 3}{5} \][/tex]
3. Solve this equation for \( y \):
[tex]\[ x = \frac{2y - 3}{5} \][/tex]
To solve for \( y \), we first eliminate the fraction by multiplying both sides by 5:
[tex]\[ 5x = 2y - 3 \][/tex]
4. Isolate \( y \):
[tex]\[ 5x + 3 = 2y \][/tex]
[tex]\[ y = \frac{5x + 3}{2} \][/tex]
Therefore, the inverse function is:
[tex]\[ f^{-1}(x) = \frac{5x + 3}{2} \][/tex]
Based on the given choices, the correct answer is:
D. [tex]\( f^{-1}(x) = \frac{5x + 3}{2} \)[/tex]
