https://file.aiquickdraw.com/mathbot/file/1727083696887wp3tfv7b.jpg

Given the image provided, we are working with the following functions and conditions:

- \( f(x) = x + 1 \)
- \( g(x) = 4x - 2 \)

We are asked to determine which of the provided statements are correct, where the statements involve compositions of the functions \( f \) and \( g \).

### Step 1: Find the compositions of \( f \) and \( g \).

#### 1. \( (g \circ f)(x) \):
\[
(g \circ f)(x) = g(f(x)) = g(x+1) = 4(x+1) - 2 = 4x + 4 - 2 = 4x + 2
\]

#### 2. \( (f \circ g)(x) \):
\[
(f \circ g)(x) = f(g(x)) = f(4x - 2) = (4x - 2) + 1 = 4x - 1
\]

### Step 2: Analyze the statements.

#### Statement (1):
\[
(g \circ f)(x) = 4x = 2
\]
From our calculation, we found that \( (g \circ f)(x) = 4x + 2 \), not \( 4x = 2 \). Hence, statement (1) is **false**.

#### Statement (2):
\[
(f \circ g)(x) = 4x - 1
\]
This is exactly what we calculated for \( (f \circ g)(x) \), so statement (2) is **true**.

#### Statement (3):
\[
(f \circ g)(x) \neq (g \circ f)(x)
\]
From our calculations, \( (f \circ g)(x) = 4x - 1 \) and \( (g \circ f)(x) = 4x + 2 \), so they are indeed not equal. Therefore, statement (3) is **true**.

#### Statement (4):
\[
(g \circ f)(x) = (f \circ g)(x)
\]
This contradicts our calculation, as the two compositions are not equal. Hence, statement (4) is **false**.

#### Statement (5):
\[
(f \circ g)(x) = (g \circ f)(x)
\]
This is also incorrect, as discussed in statement (3), so statement (5) is **false**.

### Step 3: Conclusion

The correct statements are (2) and (3). The correct answer is **e. (1), (2), and (3)**.

---

Would you like more details on any of the steps, or do you have further questions?

Here are five related questions you can explore:
1. What are the properties of function composition?
2. Can function composition be commutative?
3. How does the domain of composed functions affect the result?
4. How do you find the inverse of a composed function?
5. What are the real-life applications of function composition?

**Tip:** Always check the order when composing functions, as the order of composition can significantly change the result!

Diketahui f(x) = x + 1 dan g(x) = 4x - 2. Pernyataan yang benar adalah ...

Analyze the function composition of f(x) = x + 1 and g(x) = 4x - 2. Learn how to calculate (f ° g)(x) and (g ° f)(x) step by step. Ideal for students in Grades 10-12 studying algebra.

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