When we say "g composed of f," we are referring to the composition of two functions \( f \) and \( g \), denoted as \( g \circ f \). This composition means that you apply the function \( f \) first and then apply the function \( g \) to the result of \( f \).

Mathematically, the composition \( g \circ f \) is defined as:
\[
(g \circ f)(x) = g(f(x))
\]
Here's a step-by-step explanation:

1. **Start with a value \( x \)**: This value is the input to the function \( f \).
2. **Apply the function \( f \) to \( x \)**: Calculate \( f(x) \), which gives you a new value.
3. **Apply the function \( g \) to the result**: Take the value \( f(x) \) and use it as the input to the function \( g \). The result is \( g(f(x)) \).

### Example
If \( f(x) = 2x + 3 \) and \( g(x) = x^2 \), then the composition \( g \circ f \) is:
\[
(g \circ f)(x) = g(f(x)) = g(2x + 3)
\]
Now, apply \( g(x) = x^2 \) to \( 2x + 3 \):
\[
g(2x + 3) = (2x + 3)^2
\]
So, \( (g \circ f)(x) = (2x + 3)^2 \).

Would you like more details or have any questions?

### Relative Questions:
1. How do you find the domain of a composed function?
2. What happens if the functions \( f \) and \( g \) are inverses?
3. Can a composition of functions be commutative, i.e., is \( g \circ f = f \circ g \)?
4. How does composition relate to real-world applications?
5. What are the conditions for the composition of two functions to be continuous?

### Tip:
When working with function compositions, always carefully follow the order of operations, as \( g \circ f \) and \( f \circ g \) can yield different results.

Explore the concept of function composition with the example of g composed of f. Understand how to apply functions sequentially, starting with f and then g. Example provided with step-by-step calculations and real-world applications.

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