To find the inverse function of [tex]\( f(x) = e^{2x} \)[/tex], we need to follow these steps:
1. Start with the function equation:
[tex]\[ y = e^{2x} \][/tex]
2. Exchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function:
[tex]\[ x = e^{2y} \][/tex]
3. Solve for [tex]\( y \)[/tex]:
- Take the natural logarithm (ln) of both sides to eliminate the exponential function:
[tex]\[ \ln(x) = \ln(e^{2y}) \][/tex]
- The natural logarithm and the exponential function are inverses, so:
[tex]\[ \ln(x) = 2y \][/tex]
- Solve for [tex]\( y \)[/tex] by dividing both sides by 2:
[tex]\[ y = \frac{1}{2} \ln(x) \][/tex]
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = \frac{1}{2} \ln(x) \][/tex]
So the correct answer is:
[tex]\[ f^{-1}(x) = \frac{1}{2} \ln(x) \][/tex]
Among the given choices, the correct inverse function is:
[tex]\[ f^{-1}(x) = \frac{1}{2} \ln(x) \][/tex]
