*if f*(*x*) =$$ \sqrt{x} $$ and g(x)= Cubed root of (4-x) what is g composed of f and the domain


Given the functions $ f(x) = \sqrt{x} $ and $ g(x) = \sqrt[3]{4 - x} $, we need to find the composition $ (g \circ f)(x) $ and determine its domain.

### Step 1: Find $ g \circ f(x) $
The composition $ g \circ f(x) $ means applying $ f(x) $ first and then applying $ g $ to the result.

1. Start with $ f(x) = \sqrt{x} $.
2. Now, substitute $ \sqrt{x} $ into $ g(x) $:
   \[
   (g \circ f)(x) = g(f(x)) = g(\sqrt{x})
   \]
3. The function $ g(x) $ is $ g(x) = \sqrt[3]{4 - x} $. Substitute $ \sqrt{x} $ for $ x $ in $ g(x) $:
   \[
   (g \circ f)(x) = \sqrt[3]{4 - \sqrt{x}}
   \]

So, the composition $ g \circ f(x) = \sqrt[3]{4 - \sqrt{x}} $.

### Step 2: Determine the Domain of $ g \circ f(x) $
The domain of $ g \circ f(x) $ is the set of all $ x $ values for which $ g(f(x)) $ is defined.

1. **Domain of $ f(x) = \sqrt{x} $:** 
   \[
   x \geq 0
   \]
   This is because the square root function is only defined for non-negative numbers.

2. **Domain of $ g(x) = \sqrt[3]{4 - x} $:**
   The cube root function $ \sqrt[3]{y} $ is defined for all real numbers $ y $, so $ 4 - \sqrt{x} $ must be a real number. However, since $ \sqrt{x} $ is non-negative, $ 4 - \sqrt{x} $ must be less than or equal to 4.

   Therefore, $ 4 - \sqrt{x} \geq 0 $, which means:
   \[
   \sqrt{x} \leq 4
   \]
   Squaring both sides gives:
   \[
   x \leq 16
   \]

3. **Intersection of Domains:**
   The domain of $ g \circ f(x) $ is where both conditions $ x \geq 0 $ and $ x \leq 16 $ hold true simultaneously.

   Thus, the domain of $ g \circ f(x) $ is:
   \[
   0 \leq x \leq 16
   \]

### Final Answer:
- The composition $ (g \circ f)(x) = \sqrt[3]{4 - \sqrt{x}} $.
- The domain of $ g \circ f(x) $ is $ [0, 16] $.

Would you like more details or have any questions?

### Relative Questions:
1. How would the composition change if $ g(x) $ were $ \sqrt{4 - x} $ instead of $ \sqrt[3]{4 - x} $?
2. What is the range of the composed function $ g \circ f(x) $?
3. How do you graph the function $ g \circ f(x) = \sqrt[3]{4 - \sqrt{x}} $?
4. How does the composition $ g \circ f(x) $ behave near the endpoints of its domain?
5. What if the function $ f(x) = \sqrt{x} $ were instead $ f(x) = x^2 $? How would that affect $ g \circ f(x) $ and its domain?

### Tip:
When finding the domain of composed functions, always consider the domain restrictions of both the inner and outer functions. This ensures that the resulting function is well-defined for all values in its domain.

Explore the composition of functions g(f(x)) where f(x) = Square Root of x and g(x) = Cube Root of (4 - x). Understand how to find the composition and determine its domain step-by-step. Suitable for advanced high school students studying functions and their compositions.

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