Let `f(x)={(2x+a",",x ge -1),(bx^(2)+3",",x lt -1):}`
and `g(x)={(x+4",",0 le x le 4),(-3x-2",",-2 lt x lt 0):}`
If `a=2 and b=3,` then the range of `g(f(x))` is

Options

A. `(-2,8]`
B. `(0, 8]`
C. `[4, 8]`
D. `[-1, 8]`

Explanation

To find the range of the function \( g(f(x)) \) given the functions \( f(x) \) and \( g(x) \), we will follow these steps:

### Step 1: Define the functions with given values of \( a \) and \( b \)

Given:
- \( a = 2 \)
- \( b = 3 \)

The function \( f(x) \) can be defined as:
\[ f(x) = \begin{cases} 2x + 2 & \text{if } x \geq -1 \\ 3x + 3 & \text{if } x < -1 \end{cases} \]

The function \( g(x) \) is defined as:
\[ g(x) = \begin{cases} x + 4 & \text{if } 0 \leq x \leq 4 \\ -3x - 2 & \text{if } -2 < x < 0 \end{cases} \]

### Step 2: Find the range of \( f(x) \)

1. For \( x \geq -1 \):
\[ f(x) = 2x + 2 \]
When \( x = -1 \):
\[ f(-1) = 2(-1) + 2 = 0 \]
As \( x \) approaches \( \infty \), \( f(x) \) approaches \( \infty \). Therefore, the range for this part is \( [0, \infty) \).

2. For \( x < -1 \):
\[ f(x) = 3x + 3 \]
When \( x \) approaches \( -1 \):
\[ f(-1) = 3(-1) + 3 = 0 \]
As \( x \) approaches \( -\infty \), \( f(x) \) approaches \( -\infty \). Therefore, the range for this part is \( (-\infty, 0) \).

Combining both parts, the overall range of \( f(x) \) is:
\[ (-\infty, 0) \cup [0, \infty) = \mathbb{R} \]

### Step 3: Find the range of \( g(f(x)) \)

Now we need to evaluate \( g(f(x)) \) based on the range of \( f(x) \).

1. Since \( f(x) \) can take any real value, we need to evaluate \( g(x) \) for the ranges:
- For \( x \geq 0 \) (from \( f(x) \)):
\[ g(x) = x + 4 \]
The minimum value occurs at \( x = 0 \):
\[ g(0) = 0 + 4 = 4 \]
As \( x \) approaches \( \infty \), \( g(x) \) approaches \( \infty \). Thus, the range from this part is \( [4, \infty) \).

2. For \( x < 0 \) (from \( f(x) \)):
\[ g(x) = -3x - 2 \]
As \( x \) approaches \( 0 \) from the left:
\[ g(0) = -3(0) - 2 = -2 \]
As \( x \) approaches \( -\infty \):
\[ g(x) \to \infty \]
Thus, the range from this part is \( (-2, \infty) \).

### Step 4: Combine the ranges

Combining the ranges from both parts:
- From \( g(f(x)) \) when \( f(x) \geq 0 \): \( [4, \infty) \)
- From \( g(f(x)) \) when \( f(x) < 0 \): \( (-2, \infty) \)

The overall range of \( g(f(x)) \) is:
\[ (-2, \infty) \]

### Final Answer

The range of \( g(f(x)) \) is \( (-2, \infty) \).

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