This question is asking for the union of the ranges of two functions, $f(x)$ and $g(x)$, given their respective domains and formulas.

• The function $f(x) = x - 1$ has the domain $[2, +\infty)$. Since $f(x) = x - 1$, the range of $f(x)$ is $[1, +\infty)$, because for $x \geq 2$, subtracting 1 gives values $f(x) \geq 1$.

• The function $g(x) = \frac{1}{x^2 + 3}$ has the domain $(-\infty, 3)$, meaning it is defined for all $x$ less than 3. The expression $x^2 + 3$ is always positive and grows as $|x|$ increases. Hence, $g(x)$ will always give positive values but less than 1. Specifically, as $x \to 3^-$, the function approaches $\frac{1}{12}$, and as $x \to -\infty$, $g(x)$ approaches 0. Therefore, the range of $g(x)$ is $(0, \frac{1}{3}]$.

To find the union of the ranges of $f(x)$ and $g(x)$:

• The range of $f(x)$ is $[1, +\infty)$.
• The range of $g(x)$ is $(0, \frac{1}{3}]$.

The union of these ranges is $(0, +\infty)$, since $f(x)$ covers values greater than or equal to 1, and $g(x)$ covers values strictly between 0 and $\frac{1}{3}$.

The correct answer is option (2): $(0, +\infty)$.

Would you like more details or further clarifications?

Here are some questions to deepen your understanding:

1. What happens to the range of $g(x)$ if its domain were extended beyond $(-\infty, 3)$?
2. How would the range of $f(x) = x - 1$ change if its domain was reduced to $[0, 2]$?
3. Can you find a different function that shares the same range as $g(x)$?
4. How would you describe the behavior of $g(x)$ as $x \to 3^-$?
5. What would be the union of ranges if both functions had the same domain?

Tip: When finding the range of a function, it can help to analyze the behavior as $x \to \infty$ or as $x \to -\infty$, as well as checking for critical points within the domain.