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Sagot :
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]
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]
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