Express the function

h(x)=1x+4ℎ(�)=1�+4 in the form f∘g�∘�.

If g(x)=(x+4)�(�)=(�+4), find the function f(x)�(�).

f(x)=�(�)=

To express the function \( h(x) = 1 \cdot x + 4 \) in the form \( (f \circ g)(x) = f(g(x)) \), we need to find functions \( f \) and \( g \) such that:

1. \( g(x) = x + 4 \)
2. \( f(g(x)) = h(x) \)

Given \( g(x) = x + 4 \), we want \( f(g(x)) = h(x) = x + 4 \).

Let's break this down:

1. From the problem statement, \( g(x) = x + 4 \).
2. \( h(x) = 1 \cdot x + 4 = x + 4 \).

We can see that \( h(x) = g(x) \), meaning \( f(x) = x \).

Thus, we have:

- \( g(x) = x + 4 \)
- \( f(x) = x \)

### Answer:
The function \( f(x) \) is:

\[
f(x) = x
\]

### Questions to Expand Understanding:
1. How would the functions \( f(x) \) and \( g(x) \) change if \( h(x) \) were \( 2x + 4 \)?
2. How can the composition \( f(g(x)) \) be used to transform graphs of functions?
3. What happens if we switch the roles of \( f \) and \( g \) in the composition?
4. How would you verify that the composition \( f(g(x)) \) equals \( h(x) \)?
5. What are some real-world applications of function composition?

### Tip:
When finding functions \( f \) and \( g \) for composition \( (f \circ g)(x) \), try breaking down the expression inside the function to match the form of \( g(x) \) and then determine what \( f \) must do to \( g(x) \) to produce \( h(x) \).

Explore how to express the function h(x) = x + 4 in the form f(g(x)) using function composition. Learn about the roles of f(x) and g(x) in composing functions and their applications in mathematics.

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