If `f(x)=e^(x) and g(x)= log_(e)x`, hen which of the following is true?

Options

A. `f{g(x)} ={f(x)}`
B. `f{g(x)}=g{f(x)}`
C. `f{g(x)}+g({f(x)}=0`
D. `f{g(x)}-g{f(x)}=1`

Answer

To solve the problem, we need to evaluate \( f(g(x)) \) and \( g(f(x)) \) for the given functions \( f(x) = e^x \) and \( g(x) = \log_e x \).

### Step-by-Step Solution:

1. **Find \( f(g(x)) \)**:
- We start with \( g(x) = \log_e x \).
- Now, substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f(\log_e x) = e^{\log_e x} \]
- By the property of logarithms, \( e^{\log_e x} = x \).
- Therefore, we have:
\[ f(g(x)) = x \]

2. **Find \( g(f(x)) \)**:
- Next, we substitute \( f(x) \) into \( g(x) \):
\[ g(f(x)) = g(e^x) = \log_e(e^x) \]
- Again, using the property of logarithms, \( \log_e(e^x) = x \).
- Thus, we find:
\[ g(f(x)) = x \]

3. **Compare the results**:
- From our calculations, we have:
\[ f(g(x)) = x \quad \text{and} \quad g(f(x)) = x \]
- This implies that:
\[ f(g(x)) = g(f(x)) \]

4. **Conclusion**:
- Since both \( f(g(x)) \) and \( g(f(x)) \) yield the same result, we conclude:
\[ f(g(x)) = g(f(x)) \]
- Therefore, the correct statement is that \( f(g(x)) = g(f(x)) \).

### Final Answer:
The correct option is that \( f(g(x)) = g(f(x)) \).

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