Question

If \(f(x)=e^x\) and \(g(x)=log_{e}{x}=lnx  \) then \((gof)(x) \) is

• A. \(e^x\)
• B. \(x\)
• C. \(e^{2x}\)
• D. \(log_e{2x}\)

Answer 1

The Correct Option is B

To find \( (gof)(x) \), which means \( g(f(x)) \), we follow these steps:

1. We start with the function \( f(x)=e^x \).

2. Determine \( f(x) \): Since \( f(x) = e^x \), we substitute into the function \( g(x) = \ln x \) to get \( g(f(x)) = g(e^x) \).

3. Now, substitute \( e^x \) into \( g(x) \): \( g(e^x) = \ln(e^x) \).

4. Simplify \( \ln(e^x) \): Using the logarithmic identity \( \ln(e^x) = x \), we find \( (gof)(x) = x \).

Therefore, the result of the composition \( (gof)(x) \) is \( x \).

Option A	\( e^x \)
Option B	\( x \)
Option C	\( e^{2x} \)
Option D	\( \log_e{2x} \)

Hence, the correct answer is \( x \).