To solve this problem, we need to determine the domain and range of the composition of two functions, $f(g(x))$. This process involves understanding the domains and ranges of both functions, $f(x)$ and $g(x)$, and then finding the valid inputs for the composition.

Let's go through the steps to answer this:

(a) Domain of $f(g(x))$

The domain of the composition $f(g(x))$ is the set of all $x$ such that:

1. $x$ is in the domain of $g(x)$.
2. $g(x)$ is in the domain of $f(x)$.

Thus, we need to:

• Find the domain of $g(x)$, i.e., the values $x$ can take such that $g(x)$ is defined.
• Check if the output of $g(x)$ falls within the domain of $f(x)$.

(b) Range of $f(g(x))$

The range of $f(g(x))$ is the set of all possible outputs of the composition, i.e., the set of values that $f(g(x))$ can take.

To find this:

• Determine the range of $g(x)$, i.e., the values $g(x)$ can take.
• Plug these values into $f(x)$, and find the range of $f(x)$ based on those values.

Information Needed

It seems like the figure defining $f(x)$ and $g(x)$ is missing from the input provided. Could you upload the figure or describe the functions $f(x)$ and $g(x)$? That way, I can accurately find the domain and range of $f(g(x))$ for you!

Sample Relative Questions:

1. What are the specific values of $f(x)$ and $g(x)$ in the figure?
2. Is there any restriction on the domain or range of $f(x)$ and $g(x)$?
3. Are these functions defined piecewise, or are they continuous over their domains?
4. Do $f(x)$ and $g(x)$ have any undefined points or asymptotes?
5. Does the composition $f(g(x))$ result in any particular restrictions on the domain?

Tip: When dealing with composition of functions, always check the domain of the inner function and ensure its output fits into the domain of the outer function.