We are given the function $h(x) = -(x+5)^3 + 2$, and we are asked to find two functions $f$ and $g$ such that $(f \circ g)(x) = h(x)$.

This means we need to express $h(x)$ as a composition of two functions, i.e., $f(g(x)) = h(x)$.

Step-by-step breakdown:

1. Look at the inner structure of $h(x)$: $h(x) = -(x+5)^3 + 2$ This can be decomposed by separating the different operations being performed on $x$.

2. Define $g(x)$: We notice that the first operation inside the parentheses is adding 5 to $x$. So, a good candidate for $g(x)$ would be: $g(x) = x + 5$

3. Define $f(x)$: After applying $g(x)$, the remaining part is taking the cube of the result, multiplying by -1, and then adding 2. So, we define: $f(x) = -x^3 + 2$

Check the composition:

Now, let's check if $f(g(x))$ gives us $h(x)$.

• $g(x) = x + 5$
• $f(g(x)) = -(g(x))^3 + 2 = -(x + 5)^3 + 2$

This is indeed the original function $h(x)$.

Thus, the functions are: $g(x) = x + 5 \quad \text{and} \quad f(x) = -x^3 + 2$

Would you like further clarification or more details on the steps?

Related Questions:

1. How do we determine the inner and outer functions in a composition of functions?
2. What are other methods to break down complex functions into compositions?
3. Can we find multiple pairs of functions $f$ and $g$ for the same $h(x)$?
4. How do compositions of functions apply in real-world problems?
5. How can we verify if a composition of functions is correct?

Tip:

Always try to simplify the expression first when decomposing a function into components. Look for natural groupings of operations to identify inner and outer functions.