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Let's solve the problem step-by-step.

### Given Functions:
1. \( f(x) = \sqrt{x - 3} \)
2. \( g(x) = 2 - 4x \)

We need to find \( (f \circ g)(x) \), which represents the composite function \( f(g(x)) \), and then find the domain of this composite function.

### Step 1: Finding \( f(g(x)) \)
The composite function \( f(g(x)) \) is defined as:
\[
f(g(x)) = \sqrt{g(x) - 3}
\]

Substitute \( g(x) = 2 - 4x \) into \( f(x) \):
\[
f(g(x)) = \sqrt{(2 - 4x) - 3}
\]
\[
f(g(x)) = \sqrt{-4x - 1}
\]

### Step 2: Finding the Domain of \( f(g(x)) \)
For the square root to be defined, the expression inside the square root must be non-negative:
\[
-4x - 1 \geq 0
\]
Solve for \( x \):
\[
-4x \geq 1
\]
\[
x \leq -\frac{1}{4}
\]

Thus, the domain of \( f(g(x)) \) is all \( x \) values for which \( x \leq -\frac{1}{4} \).

### Final Answers:
1. **The composite function \( (f \circ g)(x) \):**  
   \[
   f(g(x)) = \sqrt{-4x - 1}
   \]

2. **The domain of \( f \circ g \):**  
   \[
   \left( -\infty, -\frac{1}{4} \right]
   \]

Would you like any further clarification on this problem? 

#### Related Questions:
1. How do you determine the domain of a composite function in general?
2. Why is the domain of \( f(g(x)) \) restricted to \( x \leq -\frac{1}{4} \)?
3. What would happen to the domain if the functions \( f(x) \) or \( g(x) \) were different?
4. Can \( f \circ g \) be defined if \( g(x) \) outputs a value not in the domain of \( f \)?
5. How does the composition of functions work if one function is not one-to-one?

#### Tip:
When finding the domain of a composite function \( f(g(x)) \), always ensure that the output of \( g(x) \) lies within the domain of \( f(x) \).

The functions f and g are given. Evaluate f ◦ g and find the domain of the composite function f ◦ g. f(x) = √(x - 3), g(x) = 2 - 4x.

Learn how to evaluate the composite function f ◦ g where f(x) = √(x - 3) and g(x) = 2 - 4x, and find its domain. This problem covers key concepts of composite functions and square root functions. Suitable for students in Grades 10-12.

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