AP Precalculus Practice Test: Unit 2 FRQ #1 Composite Functions, Inverse Functions, Tables

My AP Precalculus Practice Tests are carefully designed to help students build confidence for in-class assessments, support their work on AP Classroom assignments, and thoroughly prepare them for the AP Precalculus exam in May.

### AP Precalculus Practice Test: Unit 2 FRQ #1 - Composite Functions, Inverse Functions, and Tables

Given two functions \( f(x) \) and \( g(x) \) represented in tables, the tasks typically involve working with composite functions and inverses.

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### 1. **Composite Functions**:
The composite of two functions \( f \) and \( g \), written as \( (f \circ g)(x) = f(g(x)) \), means applying \( g(x) \) first, then substituting the result into \( f(x) \).

### 2. **Inverse Functions**:
The inverse function \( f^{-1}(x) \) reverses the operation of \( f(x) \). It maps outputs of \( f(x) \) back to inputs.

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### Example Problem

**Given Tables:**

**For \( f(x) \):**

| \( x \) | \( f(x) \) |
|--------|------------|
| 1 | 3 |
| 2 | 5 |
| 3 | 7 |
| 4 | 9 |

**For \( g(x) \):**

| \( x \) | \( g(x) \) |
|--------|------------|
| 1 | 4 |
| 2 | 6 |
| 3 | 8 |
| 4 | 10 |

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### Tasks:

1. **Find \( (f \circ g)(2) \)**:
- First, find \( g(2) = 6 \), but \( f(6) \) is not in the table, so \( (f \circ g)(2) \) cannot be evaluated.

2. **Find \( f^{-1}(x) \)**:
- Reverse the roles of \( x \) and \( f(x) \) in the table for \( f(x) \):

| \( f(x) \) | \( f^{-1}(x) \) |
|------------|-----------------|
| 3 | 1 |
| 5 | 2 |
| 7 | 3 |
| 9 | 4 |

3. **Find \( g^{-1}(x) \)**:
- Reverse the roles of \( x \) and \( g(x) \) in the table for \( g(x) \):

| \( g(x) \) | \( g^{-1}(x) \) |
|------------|-----------------|
| 4 | 1 |
| 6 | 2 |
| 8 | 3 |
| 10 | 4 |

4. **Verify if \( f(g(x)) \) and \( g(f(x)) \) are inverses**:
- Check if applying \( f(g(x)) \) and \( g(f(x)) \) return \( x \). In this case, \( f(g(1)) \) and \( g(f(1)) \) don’t return \( 1 \), so they are not inverses.

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### Summary:
- **Composite Function**: Apply one function to the result of another, \( f(g(x)) \).
- **Inverse Functions**: Swap the \( x \) and \( f(x) \) values in the table to find the inverse.
- **Verification**: Inverses should satisfy \( f(g(x)) = x \) and \( g(f(x)) = x \).

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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