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Let's break down the problem and solve the questions based on the provided graph and functions:

### Given Functions:
1. \( f(x) = x + 1 \)
2. \( g(x) = \sqrt{x + 3} \)
3. \( (f + g)(x) = f(x) + g(x) = x + 1 + \sqrt{x + 3} \)

### Part (a): Find \( (f + g)(-2) \) and \( (f + g)(1) \)

#### Step 1: Calculate \( (f + g)(-2) \)
We substitute \( x = -2 \) into the equation for \( (f + g)(x) \):

\[
(f + g)(-2) = (-2 + 1) + \sqrt{-2 + 3} = -1 + \sqrt{1} = -1 + 1 = 0
\]

So, \( (f + g)(-2) = 0 \).

#### Step 2: Calculate \( (f + g)(1) \)
We substitute \( x = 1 \) into the equation for \( (f + g)(x) \):

\[
(f + g)(1) = (1 + 1) + \sqrt{1 + 3} = 2 + \sqrt{4} = 2 + 2 = 4
\]

So, \( (f + g)(1) = 4 \).

### Part (b): Can you find \( (f + g)(-4) \)?

To calculate \( (f + g)(-4) \), substitute \( x = -4 \) into the equation:

\[
(f + g)(-4) = (-4 + 1) + \sqrt{-4 + 3} = -3 + \sqrt{-1}
\]

Since the square root of a negative number is not a real number, \( (f + g)(-4) \) is **undefined** in the real number system.

### Part (c): Find the domain of \( f \), \( g \), and \( f + g \)

#### Domain of \( f(x) = x + 1 \)
The function \( f(x) = x + 1 \) is a linear function, which is defined for all real numbers. So, the domain of \( f \) is:
\[
\text{Domain of } f: (-\infty, \infty)
\]

#### Domain of \( g(x) = \sqrt{x + 3} \)
For the square root function \( g(x) = \sqrt{x + 3} \), the argument inside the square root must be non-negative:
\[
x + 3 \geq 0 \quad \Rightarrow \quad x \geq -3
\]
So, the domain of \( g \) is:
\[
\text{Domain of } g: [-3, \infty)
\]

#### Domain of \( (f + g)(x) \)
Since \( (f + g)(x) = f(x) + g(x) \), the domain of \( f + g \) is the intersection of the domains of \( f \) and \( g \). Since \( f(x) \) is defined for all \( x \), the domain of \( f + g \) is the same as the domain of \( g \):
\[
\text{Domain of } (f + g): [-3, \infty)
\]

### Summary of Results:
- \( (f + g)(-2) = 0 \)
- \( (f + g)(1) = 4 \)
- \( (f + g)(-4) \) is undefined.
- Domain of \( f \): \( (-\infty, \infty) \)
- Domain of \( g \): \( [-3, \infty) \)
- Domain of \( f + g \): \( [-3, \infty) \)

---

Would you like further clarification on any part? Here are some related questions to consider:

1. What happens if you add another function \( h(x) \) to \( f + g \)?
2. Can you find \( (f \cdot g)(x) \) for \( x = 1 \)?
3. What is the range of \( g(x) \)?
4. Can \( f(x) \) and \( g(x) \) intersect, and if so, where?
5. How would you solve \( f(x) = g(x) \)?

**Tip:** Always check the domain of each function when combining them to avoid undefined expressions.

Find (f + g)(-2), (f + g)(1), (f + g)(-4), and the domain of f, g, and f + g, based on the given functions f(x) = x + 1 and g(x) = sqrt(x + 3).

Learn how to find (f + g)(-2), (f + g)(1), (f + g)(-4), and the domain of f, g, and f + g using the functions f(x) = x + 1 and g(x) = sqrt(x + 3). This problem covers algebraic functions and domain analysis, suitable for students in Grades 9-11.

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