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.