The composition of functtions is associative

Explanation

To determine whether the composition of functions is associative, we will analyze the functions and their compositions step by step.

### Step 1: Define the Functions
Let:
- \( f(x) = x \)
- \( g(x) = x + 1 \)
- \( h(x) = 2x - 1 \)

### Step 2: Understand Associative Property
The associative property states that for any three functions \( f, g, \) and \( h \):
\[ f(g(h(x))) = (f(g))(h(x)) \]
This means that the way in which the functions are grouped does not affect the result of the composition.

### Step 3: Calculate \( f(g(h(x))) \)
First, we will calculate \( f(g(h(x))) \):
1. Start with \( h(x) \):
\[ h(x) = 2x - 1 \]
2. Now apply \( g \) to \( h(x) \):
\[ g(h(x)) = g(2x - 1) = (2x - 1) + 1 = 2x \]
3. Next, apply \( f \) to \( g(h(x)) \):
\[ f(g(h(x))) = f(2x) = 2x \]

### Step 4: Calculate \( (f(g))(h(x)) \)
Now, we will calculate \( (f(g))(h(x)) \):
1. First, calculate \( f(g(x)) \):
\[ f(g(x)) = f(x + 1) = x + 1 \]
2. Now apply this result to \( h(x) \):
\[ (f(g))(h(x)) = (f(g))(2x - 1) = (2x - 1) + 1 = 2x \]

### Step 5: Compare the Results
We have:
- \( f(g(h(x))) = 2x \)
- \( (f(g))(h(x)) = 2x \)

Since both expressions are equal, we conclude that:
\[ f(g(h(x))) = (f(g))(h(x)) \]

### Conclusion
The composition of functions is associative. Therefore, the statement is **true**.

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