If `f(x) = ln(x^2 - x + 2) ; RR^+ rarr RR` and `g(x) = {x} + 1; [1, 2] rarr [1, 2],` where `{x}` denotes fractional part of `x`.Find the domain and range of `f(g(x))` when defined.

Explanation

To find the domain and range of the composite function \( f(g(x)) \) where \( f(x) = \ln(x^2 - x + 2) \) and \( g(x) = \{x\} + 1 \), we will follow these steps:

### Step 1: Determine the range of \( g(x) \)

The function \( g(x) = \{x\} + 1 \) represents the fractional part of \( x \) plus 1. The fractional part \( \{x\} \) takes values in the interval \([0, 1)\). Therefore, adding 1 to this gives us:

\[ g(x) \in [1, 2) \]

### Step 2: Identify the domain of \( f(x) \)

Next, we need to find the domain of \( f(x) = \ln(x^2 - x + 2) \). The argument of the logarithm must be positive:

\[ x^2 - x + 2 > 0 \]

To analyze this quadratic expression, we can find its discriminant:

\[ D = b^2 - 4ac = (-1)^2 - 4 \cdot 1 \cdot 2 = 1 - 8 = -7 \]

Since the discriminant is negative, the quadratic \( x^2 - x + 2 \) does not have real roots and is always positive for all \( x \in \mathbb{R} \). Thus, the domain of \( f(x) \) is:

\[ \text{Domain of } f(x) = \mathbb{R} \]

### Step 3: Determine the domain of \( f(g(x)) \)

The domain of the composite function \( f(g(x)) \) is determined by the range of \( g(x) \) and the domain of \( f(x) \). Since \( g(x) \) takes values in \([1, 2)\) and \( f(x) \) is defined for all real numbers, the domain of \( f(g(x)) \) is:

\[ \text{Domain of } f(g(x)) = [1, 2) \]

### Step 4: Find the range of \( f(g(x)) \)

Now we need to find the range of \( f(g(x)) \). We substitute \( g(x) \) into \( f(x) \):

\[ f(g(x)) = f(\{x\} + 1) = \ln((\{x\} + 1)^2 - (\{x\} + 1) + 2) \]

Simplifying the expression inside the logarithm:

\[ (\{x\} + 1)^2 - (\{x\} + 1) + 2 = \{x\}^2 + 2\{x\} + 1 - \{x\} - 1 + 2 = \{x\}^2 + \{x\} + 2 \]

Thus, we have:

\[ f(g(x)) = \ln(\{x\}^2 + \{x\} + 2) \]

### Step 5: Analyze the expression \( \{x\}^2 + \{x\} + 2 \)

The fractional part \( \{x\} \) varies from \( 0 \) to \( 1 \). Therefore, we can evaluate the minimum and maximum values of \( \{x\}^2 + \{x\} + 2 \):

- When \( \{x\} = 0 \):
\[ 0^2 + 0 + 2 = 2 \]
- When \( \{x\} = 1 \):
\[ 1^2 + 1 + 2 = 4 \]

Thus, the expression \( \{x\}^2 + \{x\} + 2 \) takes values in the interval \([2, 4]\).

### Step 6: Find the range of \( f(g(x)) \)

Now we take the logarithm of the interval \([2, 4]\):

\[ \text{Range of } f(g(x)) = [\ln(2), \ln(4)] \]

Since \( \ln(4) = 2\ln(2) \), we can express the range as:

\[ \text{Range of } f(g(x)) = [\ln(2), 2\ln(2)] \]

### Final Answer

- **Domain of \( f(g(x)) \)**: \([1, 2)\)
- **Range of \( f(g(x)) \)**: \([\ln(2), 2\ln(2)]\)