Answer :
To find \((f \bullet f^{-1})(3)\), we need to follow a few steps:
1. Determine the inverse function \(f^{-1}(x)\) of \(f(x) = 3x\):
- Given the function \(f(x) = 3x\), we want to find the inverse function \(f^{-1}(x)\).
- To find the inverse, we start with the equation \(y = 3x\), and solve for \(x\):
[tex]\[ y = 3x \implies x = \frac{y}{3} \][/tex]
- Therefore, the inverse function is \(f^{-1}(x) = \frac{x}{3}\).
2. Calculate \(f^{-1}(3)\):
- Using the inverse function \(f^{-1}(x) = \frac{x}{3}\), we substitute \(x = 3\):
[tex]\[ f^{-1}(3) = \frac{3}{3} = 1 \][/tex]
3. Calculate \(f(f^{-1}(3))\):
- We found that \(f^{-1}(3) = 1\). Now, substitute this back into the original function \(f(x) = 3x\):
[tex]\[ f(f^{-1}(3)) = f(1) = 3 \cdot 1 = 3 \][/tex]
Thus, \((f \bullet f^{-1})(3) = 3\).
The correct answer is:
C. [tex]\( \boxed{3} \)[/tex]
1. Determine the inverse function \(f^{-1}(x)\) of \(f(x) = 3x\):
- Given the function \(f(x) = 3x\), we want to find the inverse function \(f^{-1}(x)\).
- To find the inverse, we start with the equation \(y = 3x\), and solve for \(x\):
[tex]\[ y = 3x \implies x = \frac{y}{3} \][/tex]
- Therefore, the inverse function is \(f^{-1}(x) = \frac{x}{3}\).
2. Calculate \(f^{-1}(3)\):
- Using the inverse function \(f^{-1}(x) = \frac{x}{3}\), we substitute \(x = 3\):
[tex]\[ f^{-1}(3) = \frac{3}{3} = 1 \][/tex]
3. Calculate \(f(f^{-1}(3))\):
- We found that \(f^{-1}(3) = 1\). Now, substitute this back into the original function \(f(x) = 3x\):
[tex]\[ f(f^{-1}(3)) = f(1) = 3 \cdot 1 = 3 \][/tex]
Thus, \((f \bullet f^{-1})(3) = 3\).
The correct answer is:
C. [tex]\( \boxed{3} \)[/tex]
