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 $h(x)$.