*Which of the following are True about the notation f(g(x))�(�(�)). Check all that apply.*

The output of one function is used as the input for another function

f(x)�(�) is used as the input into g(x)�(�)

g(x)�(�) is used as the input into f(x)�(�)

The process is called function composition

Let's analyze each statement about the notation \( f(g(x)) \), which is a common way to represent the composition of two functions \( f \) and \( g \):

1. **The output of one function is used as the input for another function.**
   - **True.** In \( f(g(x)) \), the function \( g(x) \) is computed first, and its output is then used as the input to the function \( f \). This is the essence of function composition.

2. **\( f(x) \) is used as the input into \( g(x) \).**
   - **False.** In \( f(g(x)) \), it is not \( f(x) \) being used as input to \( g(x) \). Instead, \( g(x) \) is used as the input to \( f \). 

3. **\( g(x) \) is used as the input into \( f(x) \).**
   - **True.** This is precisely what the notation \( f(g(x)) \) means: the output of \( g(x) \) is used as the input for \( f \).

4. **The process is called function composition.**
   - **True.** The notation \( f(g(x)) \) represents the composition of two functions, which is the process of applying one function to the results of another.

### Summary of True Statements:
1. The output of one function is used as the input for another function.
2. \( g(x) \) is used as the input into \( f(x) \).
3. The process is called function composition.

### Questions to Expand Understanding:
1. What is the difference between \( f(g(x)) \) and \( g(f(x)) \)?
2. How would you represent the composition of three functions \( f \), \( g \), and \( h \)?
3. What happens if \( g(x) = x^2 \) and \( f(x) = x + 1 \), and you compute \( f(g(x)) \)?
4. How can function composition be useful in real-world applications like computer science or physics?
5. Is function composition commutative? That is, does \( f(g(x)) = g(f(x)) \) hold in general?

### Tip:
When composing functions, always evaluate from the inside out, starting with the innermost function. This approach helps in understanding the flow of data through the composition.

Explore the concept of function composition with examples. Learn how f(g(x)) represents the composition of functions, where g(x) is used as input into f(x). Understand the significance of function composition in mathematics and its real-world applications.

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