`f(x) = |x-1|, f: R^+->R, g(x) = e^x, g:[-1,oo)->R`. If the function fog(x) is defined, then it domain and range respectively are:

Options

A. `(0,oo)` and `[0,oo)`
B. `[-1,oo)` and `[0,oo)`
C. `[-1,oo)` and `[1-1/e-oo)`
D. `[-1,oo)` and `[1/3-1,oo)`

Explanation

To solve the problem, we need to find the domain and range of the composite function \( f(g(x)) \) where \( f(x) = |x - 1| \) and \( g(x) = e^x \).

### Step-by-Step Solution:

1. **Identify the Functions and Their Domains**:
- The function \( f(x) = |x - 1| \) is defined for all real numbers \( x \). Thus, the domain of \( f \) is \( \mathbb{R} \).
- The function \( g(x) = e^x \) is defined for \( x \in [-1, \infty) \). Thus, the domain of \( g \) is \( [-1, \infty) \).

2. **Determine the Composite Function**:
- The composite function \( f(g(x)) \) can be expressed as:
\[ f(g(x)) = f(e^x) = |e^x - 1| \]

3. **Find the Domain of \( f(g(x)) \)**:
- Since \( g(x) \) is defined for \( x \in [-1, \infty) \), we need to check if \( g(x) \) outputs values that are in the domain of \( f \).
- The output of \( g(x) = e^x \) for \( x \in [-1, \infty) \) is:
- At \( x = -1 \), \( g(-1) = e^{-1} \approx 0.3679 \).
- As \( x \) approaches \( \infty \), \( g(x) \) approaches \( \infty \).
- Therefore, the range of \( g(x) \) is \( (e^{-1}, \infty) \) which is approximately \( (0.3679, \infty) \).
- Since \( f(x) \) is defined for all real numbers, the domain of \( f(g(x)) \) is the same as the domain of \( g(x) \), which is \( [-1, \infty) \).

4. **Find the Range of \( f(g(x)) \)**:
- To find the range of \( f(g(x)) = |e^x - 1| \):
- When \( e^x < 1 \) (which happens for \( x < 0 \)), \( |e^x - 1| = 1 - e^x \).
- When \( e^x \geq 1 \) (which happens for \( x \geq 0 \)), \( |e^x - 1| = e^x - 1 \).
- For \( x < 0 \):
- As \( x \) approaches \( -1 \), \( e^{-1} \approx 0.3679 \) and \( |e^{-1} - 1| \approx 0.6321 \).
- As \( x \) approaches \( 0 \), \( |e^0 - 1| = 0 \).
- Therefore, the minimum value is \( 0 \) and the maximum value approaches \( 1 \) from below.
- For \( x \geq 0 \):
- As \( x \) approaches \( 0 \), \( |e^0 - 1| = 0 \).
- As \( x \) approaches \( \infty \), \( |e^x - 1| \) approaches \( \infty \).
- Thus, the range of \( f(g(x)) \) is \( [0, \infty) \).

### Final Answer:
- **Domain of \( f(g(x)) \)**: \( [-1, \infty) \)
- **Range of \( f(g(x)) \)**: \( [0, \infty) \)