Find the composite functions (f ∘ g) and (g ∘ f). What is the domain of each composite function? (Simplify your answers completely. Enter your domains using interval notation.)
f(x) = x4
g(x) = 
4sqrt4


 
(f ∘ g)(x)
=	
 
domain of	
(f ∘ g)(x)
 
(g ∘ f)(x)
=	
 
domain of	
(g ∘ f)(x)


To find the composite functions \( (f \circ g)(x) \) and \( (g \circ f)(x) \), let's first define the given functions:

- \( f(x) = x^4 \)
- \( g(x) = \sqrt[4]{x} \)

Now, let's go through each composite function step by step:

### 1. Composite function \( (f \circ g)(x) \)

This means we apply \( g(x) \) first and then apply \( f \) to the result:

\[
(f \circ g)(x) = f(g(x))
\]

Given \( g(x) = \sqrt[4]{x} \), substitute this into \( f \):

\[
f(g(x)) = f\left(\sqrt[4]{x}\right) = \left(\sqrt[4]{x}\right)^4
\]

Since raising a fourth root to the power of 4 cancels out the root:

\[
(f \circ g)(x) = x
\]

#### Domain of \( (f \circ g)(x) \):
For \( g(x) = \sqrt[4]{x} \), the domain is \( x \geq 0 \), because the fourth root of a negative number is not a real number. Since \( f(x) = x^4 \) is defined for all real numbers, the domain of the composite function \( (f \circ g)(x) = x \) is the domain of \( g(x) \), which is \( [0, \infty) \).

### 2. Composite function \( (g \circ f)(x) \)

This means we apply \( f(x) \) first and then apply \( g \) to the result:

\[
(g \circ f)(x) = g(f(x))
\]

Given \( f(x) = x^4 \), substitute this into \( g \):

\[
g(f(x)) = g\left(x^4\right) = \sqrt[4]{x^4}
\]

Again, taking the fourth root of \( x^4 \):

\[
(g \circ f)(x) = |x|
\]

The absolute value arises because for \( x^4 \), the fourth power is always positive, and when we take the fourth root, we need to consider both the positive and negative values of \( x \).

#### Domain of \( (g \circ f)(x) \):
Since \( f(x) = x^4 \) is defined for all real numbers and \( g(x) = \sqrt[4]{x} \) is defined for non-negative numbers, the domain of the composite function \( (g \circ f)(x) = |x| \) is all real numbers \( (-\infty, \infty) \).

### Final Answers:

1. \( (f \circ g)(x) = x \)
   - Domain: \( [0, \infty) \)

2. \( (g \circ f)(x) = |x| \)
   - Domain: \( (-\infty, \infty) \)

Do you want more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How would the domain change if \( f(x) \) involved an even root?
2. What happens to the composite functions if \( g(x) \) is a cube root instead of a fourth root?
3. Can you generalize the process for any power function?
4. How do we handle composite functions when the domain of \( g(x) \) is restricted by \( f(x) \)?
5. How does the range of a function affect the composite functions?

**Tip**: Always check the domain restrictions of both functions when composing functions to avoid undefined expressions.

Find the composite functions (f ∘ g) and (g ∘ f). What is the domain of each composite function? (Simplify your answers completely. Enter your domains using interval notation.) f(x) = x^4, g(x) = √[4]{x}

Learn how to find the composite functions (f ∘ g)(x) and (g ∘ f)(x) and determine their domains with a detailed step-by-step solution. This problem covers key mathematical concepts, including composite functions, function operations, and domain analysis. Suitable for students in Grades 10-12.

Math Problem Statement

(f ∘ g)(x)

(g ∘ f)(x)

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