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.