1.	 F(x) = (2x - 6)^2 + 9 
	2.	 F(x) = (6x + 12)^2 
	3.	 F(x) = \frac{2x - 3}{6x - 1} 
	4.	 F(x) = \frac{x}{3 + 7x} 
	5.	 F(x) = \frac{3x + 6}{2x - 9}  tentukan invers bilangan diatas

To find the inverse of each given function \( F(x) \), we follow these general steps:

1. Replace \( F(x) \) with \( y \).
2. Swap \( x \) and \( y \).
3. Solve for \( y \).

Let's determine the inverse of each function:

### 1. \( F(x) = (2x - 6)^2 + 9 \)

**Step-by-step:**

1. Replace \( F(x) \) with \( y \):
   \[
   y = (2x - 6)^2 + 9
   \]
2. Swap \( x \) and \( y \):
   \[
   x = (2y - 6)^2 + 9
   \]
3. Solve for \( y \):
   \[
   x - 9 = (2y - 6)^2
   \]
   \[
   \sqrt{x - 9} = 2y - 6
   \]
   \[
   y = \frac{\sqrt{x - 9} + 6}{2} \quad \text{and} \quad y = \frac{-\sqrt{x - 9} + 6}{2}
   \]
   Since squaring introduces both positive and negative roots, the inverse is a piecewise function.

**Inverse:** \( F^{-1}(x) = \frac{\sqrt{x - 9} + 6}{2} \) and \( F^{-1}(x) = \frac{-\sqrt{x - 9} + 6}{2} \)

---

### 2. \( F(x) = (6x + 12)^2 \)

**Step-by-step:**

1. Replace \( F(x) \) with \( y \):
   \[
   y = (6x + 12)^2
   \]
2. Swap \( x \) and \( y \):
   \[
   x = (6y + 12)^2
   \]
3. Solve for \( y \):
   \[
   \sqrt{x} = 6y + 12
   \]
   \[
   y = \frac{\sqrt{x} - 12}{6} \quad \text{and} \quad y = \frac{-\sqrt{x} - 12}{6}
   \]

**Inverse:** \( F^{-1}(x) = \frac{\sqrt{x} - 12}{6} \) and \( F^{-1}(x) = \frac{-\sqrt{x} - 12}{6} \)

---

### 3. \( F(x) = \frac{2x - 3}{6x - 1} \)

**Step-by-step:**

1. Replace \( F(x) \) with \( y \):
   \[
   y = \frac{2x - 3}{6x - 1}
   \]
2. Swap \( x \) and \( y \):
   \[
   x = \frac{2y - 3}{6y - 1}
   \]
3. Solve for \( y \):
   \[
   x(6y - 1) = 2y - 3
   \]
   \[
   6xy - x = 2y - 3
   \]
   \[
   6xy - 2y = x - 3
   \]
   \[
   y(6x - 2) = x - 3
   \]
   \[
   y = \frac{x - 3}{6x - 2}
   \]

**Inverse:** \( F^{-1}(x) = \frac{x - 3}{6x - 2} \)

---

### 4. \( F(x) = \frac{x}{3 + 7x} \)

**Step-by-step:**

1. Replace \( F(x) \) with \( y \):
   \[
   y = \frac{x}{3 + 7x}
   \]
2. Swap \( x \) and \( y \):
   \[
   x = \frac{y}{3 + 7y}
   \]
3. Solve for \( y \):
   \[
   x(3 + 7y) = y
   \]
   \[
   3x + 7xy = y
   \]
   \[
   7xy - y = -3x
   \]
   \[
   y(7x - 1) = -3x
   \]
   \[
   y = \frac{-3x}{7x - 1}
   \]

**Inverse:** \( F^{-1}(x) = \frac{-3x}{7x - 1} \)

---

### 5. \( F(x) = \frac{3x + 6}{2x - 9} \)

**Step-by-step:**

1. Replace \( F(x) \) with \( y \):
   \[
   y = \frac{3x + 6}{2x - 9}
   \]
2. Swap \( x \) and \( y \):
   \[
   x = \frac{3y + 6}{2y - 9}
   \]
3. Solve for \( y \):
   \[
   x(2y - 9) = 3y + 6
   \]
   \[
   2xy - 9x = 3y + 6
   \]
   \[
   2xy - 3y = 9x + 6
   \]
   \[
   y(2x - 3) = 9x + 6
   \]
   \[
   y = \frac{9x + 6}{2x - 3}
   \]

**Inverse:** \( F^{-1}(x) = \frac{9x + 6}{2x - 3} \)

---

Would you like more details on any of these solutions or have any questions?

Here are 5 related questions to expand on the concept:

1. How do we determine the domain and range of an inverse function?
2. What happens to the graph of a function when you find its inverse?
3. Are there functions that do not have inverses? If so, why?
4. How can we verify that a function is the inverse of another?
5. Why do some functions produce multiple inverses when squared?

**Tip:** When finding inverses of functions, always check that the function is one-to-one (passes the horizontal line test), as this guarantees the existence of an inverse.

1. F(x) = (2x - 6)^2 + 9 
2. F(x) = (6x + 12)^2 
3. F(x) = \frac{2x - 3}{6x - 1} 
4. F(x) = \frac{x}{3 + 7x} 
5. F(x) = \frac{3x + 6}{2x - 9} 
tentukan invers bilangan diatas

This guide walks through the process of finding the inverse of five different functions, including rational and quadratic functions. Detailed solutions show how to swap x and y, solve for y, and handle square roots. Ideal for high school students studying inverse functions.

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