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Use composition to verify if the given pairs are inverses of each other

To verify if the given pairs of functions are inverses of each other, we need to check if the composition of the two functions, \( f(g(x)) \) and \( g(f(x)) \), equals \( x \) for each pair.

Let's go through each pair step by step.

### 1. Pair 1:
\[ f(x) = 3x + 6 \]
\[ g(x) = \frac{1}{3}x - 2 \]

**Step 1:** Compute \( f(g(x)) \).
\[
f(g(x)) = 3\left(\frac{1}{3}x - 2\right) + 6 = x - 6 + 6 = x
\]

**Step 2:** Compute \( g(f(x)) \).
\[
g(f(x)) = \frac{1}{3}(3x + 6) - 2 = x + 2 - 2 = x
\]

Since \( f(g(x)) = x \) and \( g(f(x)) = x \), the functions \( f(x) \) and \( g(x) \) are inverses of each other.

### 2. Pair 2:
\[ f(x) = -2x - 6 \]
\[ g(x) = \frac{-1}{2}x + 3 \]

**Step 1:** Compute \( f(g(x)) \).
\[
f(g(x)) = -2\left(\frac{-1}{2}x + 3\right) - 6 = x - 6 - 6 = x - 12
\]
\( f(g(x)) \neq x \).

**Step 2:** Compute \( g(f(x)) \).
\[
g(f(x)) = \frac{-1}{2}(-2x - 6) + 3 = x + 3 + 3 = x + 6
\]
\( g(f(x)) \neq x \).

Since neither \( f(g(x)) = x \) nor \( g(f(x)) = x \), the functions \( f(x) \) and \( g(x) \) are not inverses of each other.

### 3. Pair 3:
\[ f(x) = -2x + 4 \]
\[ g(x) = \frac{-1}{2}x + 2 \]

**Step 1:** Compute \( f(g(x)) \).
\[
f(g(x)) = -2\left(\frac{-1}{2}x + 2\right) + 4 = x - 4 + 4 = x
\]

**Step 2:** Compute \( g(f(x)) \).
\[
g(f(x)) = \frac{-1}{2}(-2x + 4) + 2 = x - 2 + 2 = x
\]

Since \( f(g(x)) = x \) and \( g(f(x)) = x \), the functions \( f(x) \) and \( g(x) \) are inverses of each other.

### 4. Pair 4:
\[ f(x) = -\frac{3}{5}x - 10 \]
\[ g(x) = \frac{-5}{3}x + 12 \]

**Step 1:** Compute \( f(g(x)) \).
\[
f(g(x)) = -\frac{3}{5}\left(\frac{-5}{3}x + 12\right) - 10 = x - 36 - 10 = x - 46
\]
\( f(g(x)) \neq x \).

**Step 2:** Compute \( g(f(x)) \).
\[
g(f(x)) = \frac{-5}{3}\left(-\frac{3}{5}x - 10\right) + 12 = x + 50 + 12 = x + 62
\]
\( g(f(x)) \neq x \).

Since neither \( f(g(x)) = x \) nor \( g(f(x)) = x \), the functions \( f(x) \) and \( g(x) \) are not inverses of each other.

### Summary:
- **Pair 1** and **Pair 3** are inverses of each other.
- **Pair 2** and **Pair 4** are not inverses of each other.

Would you like any more details on the steps or have any other questions?

Here are five related questions you might find helpful:
1. How can you algebraically find the inverse of a function?
2. What are the conditions for two functions to be inverses of each other?
3. Can a function have more than one inverse? Why or why not?
4. How can you graphically verify if two functions are inverses?
5. What happens if you compose a function with its inverse?

**Tip:** When verifying if two functions are inverses, always ensure that both compositions \( f(g(x)) \) and \( g(f(x)) \) simplify to \( x \). If either one does not, the functions are not inverses.

Learn how to verify if given pairs of functions are inverses of each other with detailed examples and step-by-step solutions. Understand the concept of inverse functions and composition of functions. Suitable for Grades 11-12.

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