If `f(x)=e^(x) and g(x)=f(x)+f^(-1)`, what does g(2) equal?

Options

A. 5.1
B. 7.4
C. 7.5
D. 8.1

Explanation

To solve the problem, we need to find \( g(2) \) given that \( f(x) = e^x \) and \( g(x) = f(x) + f^{-1}(x) \).

### Step-by-Step Solution

1. **Identify the function \( f(x) \)**:
\[ f(x) = e^x \]

2. **Find the inverse function \( f^{-1}(x) \)**:
- Start by letting \( y = f(x) \):
\[ y = e^x \]
- Swap \( x \) and \( y \):
\[ x = e^y \]
- Now, solve for \( y \):
- Take the natural logarithm of both sides:
\[ \ln(x) = \ln(e^y) \]
- Using the property of logarithms \( \ln(e^y) = y \):
\[ \ln(x) = y \]
- Thus, we have:
\[ f^{-1}(x) = \ln(x) \]

3. **Substitute \( f(x) \) and \( f^{-1}(x) \) into \( g(x) \)**:
\[ g(x) = f(x) + f^{-1}(x) = e^x + \ln(x) \]

4. **Evaluate \( g(2) \)**:
\[ g(2) = e^2 + \ln(2) \]
- Calculate \( e^2 \) (approximately 7.389) and \( \ln(2) \) (approximately 0.693):
\[ g(2) \approx 7.389 + 0.693 \approx 8.082 \]

5. **Final result**:
\[ g(2) \approx 8.1 \]

### Conclusion
Thus, \( g(2) \) equals approximately 8.1.