Two functions  and  are defined in the figure below.
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Find the domain and range of the composition . Write your answers in set notation.

(a)Domain of  : 

(b)Range of  : 


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.

Two functions f(x) and g(x) are defined in the figure below. Find the domain and range of the composition f(g(x)). Write your answers in set notation.
(a) Domain of f(g(x)):
(b) Range of f(g(x)):

Learn how to find the domain and range of the composition of two functions, f(x) and g(x). This problem explains the process step by step, focusing on function composition and the use of set notation for the domain and range. Suitable for Grades 9-12.

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