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**Analyzing the content...**The image contains a problem regarding functions and their behavior.

Problem:

It is given that $f(x) = \sqrt{3 - x}$ and $g(x) = \frac{2}{x + 1}$. We need to find the domain of the composition function $(f \circ g)(x) = f(g(x))$.

Step-by-step Solution:

1. Domain of $f(x) = \sqrt{3 - x}$:

   For the square root function to be defined, the expression inside the square root must be non-negative:

   $3 - x \geq 0$

   Solving for $x$:

   $x \leq 3$

   Thus, the domain of $f(x)$ is $(-\infty, 3]$.

2. Domain of $g(x) = \frac{2}{x + 1}$:

   For the fraction to be defined, the denominator must not be zero:

   $x + 1 \neq 0 \implies x \neq -1$

   Thus, the domain of $g(x)$ is $(-\infty, -1) \cup (-1, \infty)$.

3. Domain of the composition function $(f \circ g)(x) = f(g(x))$:

   To find the domain of the composition, we need to determine the set of all $x$ values that are in the domain of $g(x)$ such that $g(x)$ is in the domain of $f(x)$.

   Since $f(x) = \sqrt{3 - x}$, we need:

   $3 - g(x) \geq 0.$

   Substituting $g(x) = \frac{2}{x + 1}$:

   $3 - \frac{2}{x + 1} \geq 0.$

   Rearranging this inequality:

   $3(x + 1) - 2 \geq 0,$

   $3x + 3 - 2 \geq 0,$

   $3x + 1 \geq 0,$

   $x \geq -\frac{1}{3}.$

4. Combine the domain conditions:

   We also know that $x \neq -1$. Combining the conditions $x \geq -\frac{1}{3}$ and $x \neq -1$, we get the domain:

   $x \in \left[-\frac{1}{3}, \infty\right) \setminus \{-1\}.$

Conclusion:

The domain of the function $(f \circ g)(x)$ is:

$\left[-\frac{1}{3}, \infty\right) \setminus \{-1\}.$

Would you like more details on any specific step or do you have any questions?

Follow-up Questions:

1. What is the range of the composition function $(f \circ g)(x)$?
2. How would you find the domain of $(g \circ f)(x)$ instead?
3. Can you explain why the composition of two functions depends on the range of the inner function?
4. What happens to the domain if both functions $f(x)$ and $g(x)$ are modified?
5. How would you graph the composition function $(f \circ g)(x)$?

Tip:

Always carefully consider the domains of