The question in your image asks to express the given function $h(x)$ as a composition of two functions $f$ and $g$, such that $h(x) = (f \circ g)(x) = f(g(x))$.

The given function is:

$h(x) = \sqrt[5]{x^7} - 4$

Step 1: Break Down $h(x)$ into Two Functions

We need to identify two functions $f(x)$ and $g(x)$ such that: $h(x) = f(g(x))$

Let's first handle the inside part:

• The function under the radical is $x^7$. We can choose $g(x) = x^7$.

Now, we are left with $f(x)$, which should operate on $g(x)$. Since we are applying a fifth root and then subtracting 4, we define: $f(x) = \sqrt[5]{x} - 4$

Step 2: Verify

Now, compose the functions to verify: $h(x) = f(g(x)) = f(x^7) = \sqrt[5]{x^7} - 4$ This matches the given $h(x)$.

Final Answer

The functions are: $f(x) = \sqrt[5]{x} - 4$ $g(x) = x^7$

Would you like further details on this solution or have any other questions?

Here are 5 related questions to explore:

1. How do you compose two functions in general?
2. What is the difference between composing and adding functions?
3. How can we reverse a composition of functions?
4. What are some real-life examples of function composition?
5. How does the chain rule apply to compositions of functions in calculus?

Tip: When breaking down a function for composition, always start by identifying the innermost operation, which can help define the function $g(x)$.