The problem asks to find $f \circ g$ (composition of functions) and the domain, where the functions are:

1. $f(x) = x^4$
2. $g(x) = (x + 1)^{\frac{1}{2}}$ (or the square root of $x+1$)

Step 1: Find $f \circ g$ (composition of functions)

By definition, $f \circ g (x) = f(g(x))$.

Substitute $g(x) = (x + 1)^{\frac{1}{2}}$ into $f(x) = x^4$: $f(g(x)) = \left( (x + 1)^{\frac{1}{2}} \right)^4$ Simplify: $f(g(x)) = (x + 1)^2$

So, $f \circ g(x) = (x + 1)^2$.

Step 2: Determine the Domain

The domain of $f \circ g$ depends on both the domain of $g(x)$ and whether the composition makes sense.

• The function $g(x) = (x + 1)^{\frac{1}{2}}$ (square root function) is only defined when $x + 1 \geq 0$, which implies: $x \geq -1$

• The function $f(x) = x^4$ is defined for all real numbers, so the domain of the composite function $f(g(x))$ is the same as the domain of $g(x)$.

Thus, the domain of $f \circ g$ is $x \geq -1$.

Final Answer:

• $f \circ g(x) = (x + 1)^2$
• Domain: $x \geq -1$

Would you like more details on any step or have questions about related concepts?

Here are 5 related questions you might explore:

1. What is the domain of $f(g(x))$ if the functions are reversed?
2. How would the composition change if $f(x)$ or $g(x)$ were different types of functions?
3. What happens to the range of $f \circ g$ in this case?
4. How do you find the domain of a composition of trigonometric functions?
5. How does the order of function composition affect the result?

Tip: When finding the domain of a composite function, always consider the domains of both functions individually before combining them!