Consider the functions
`f(x)={(x+1",",x le 1),(2x+1",",1lt x le 2):} and g(x)={(x^(2)",", -1 le x lt2),(x+2",",2le x le 3):}`
The range of the function `f(g(x))` is

Options

A. `[1,5]`
B. `[2,3]`
C. `[1,2] cup [3,5]`
D. None of these

Explanation

To find the range of the function \( f(g(x)) \), we need to analyze the functions \( f(x) \) and \( g(x) \) given in the problem.

### Step 1: Define the functions
The functions are defined as follows:
- \( f(x) = \begin{cases} x + 1 & \text{if } x \leq 1 \\ 2x + 1 & \text{if } 1 < x \leq 2 \end{cases} \)

- \( g(x) = \begin{cases} x^2 & \text{if } -1 \leq x < 2 \\ x + 2 & \text{if } 2 \leq x \leq 3 \end{cases} \)

### Step 2: Find the range of \( g(x) \)
1. For \( -1 \leq x < 2 \):
- The function \( g(x) = x^2 \) takes values from \( g(-1) = 1 \) to \( g(2) = 4 \) (but does not include 4).
- Thus, the range of \( g(x) \) in this interval is \( [0, 4) \).

2. For \( 2 \leq x \leq 3 \):
- The function \( g(x) = x + 2 \) takes values from \( g(2) = 4 \) to \( g(3) = 5 \).
- Thus, the range of \( g(x) \) in this interval is \( [4, 5] \).

Combining both intervals, the overall range of \( g(x) \) is:
\[ [0, 5] \]

### Step 3: Find the range of \( f(g(x)) \)
Now we will analyze \( f(g(x)) \) based on the range of \( g(x) \).

1. **For \( g(x) \in [0, 1] \)**:
- Here, we use \( f(x) = x + 1 \).
- Thus, \( f(g(x)) = g(x) + 1 \) which gives us the range \( [0 + 1, 1 + 1] = [1, 2] \).

2. **For \( g(x) \in (1, 2] \)**:
- Here, we use \( f(x) = 2x + 1 \).
- Thus, \( f(g(x)) = 2g(x) + 1 \).
- The minimum value occurs at \( g(x) = 1 \): \( 2(1) + 1 = 3 \).
- The maximum value occurs at \( g(x) = 2 \): \( 2(2) + 1 = 5 \).
- Therefore, the range for this part is \( (3, 5] \).

### Step 4: Combine the ranges
Now we combine the ranges from both cases:
- From the first case, we have \( [1, 2] \).
- From the second case, we have \( (3, 5] \).

Thus, the overall range of \( f(g(x)) \) is:
\[ [1, 2] \cup (3, 5] \]

### Final Answer
The range of the function \( f(g(x)) \) is:
\[ [1, 2] \cup (3, 5] \]

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