AP Precalculus Practice Test: Unit 2 Question #22 Tables, Functions, and Inverses

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### AP Precalculus Practice Test: Unit 2, Question #22
**Topic:** Tables, Functions, and Inverses

In this question, you are asked to analyze a **table of values** and identify both the **function** and its **inverse**. The key steps involve identifying the relationship between the values in the table and then using that to find the inverse of the function.

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### **Steps to Analyze Tables, Functions, and Inverses:**

1. **Identify the function from the table:**
- In most problems, the table will provide values of \( x \) and \( f(x) \). You can use these pairs to identify the rule that describes the function. If the function is linear, exponential, or follows some other pattern, try to recognize the relationship.

2. **Check for a one-to-one relationship:**
- For a function to have an inverse, it must be **one-to-one**, meaning each \( x \)-value must correspond to exactly one \( f(x) \)-value, and each \( f(x) \)-value must correspond to exactly one \( x \)-value.

3. **Determine the inverse of the function:**
- Once the function is identified, find the inverse by swapping the \( x \)- and \( f(x) \)-values in the table. The inverse function reverses the roles of the dependent and independent variables.

4. **Create a new table for the inverse:**
- After swapping the values, you can create a new table that represents the inverse function. This new table will provide pairs of values where the first column represents the output of the inverse function, and the second column represents the input.

5. **Verify the inverse:**
- For a correct inverse, if you apply the function and then its inverse (or vice versa), you should get the original value of \( x \) back. This can be checked by looking for symmetry in the tables (the pairs should reverse when swapped).

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### **Example Problem:**

Given the following table of values for a function \( f(x) \):

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

Find the inverse of the function.

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### **Solution Steps:**

1. **Identify the function from the table:**
- The values of \( f(x) \) increase by 2 for each increase of 1 in \( x \), which suggests a linear function. The pattern appears to follow \( f(x) = 2x \).

2. **Check for a one-to-one relationship:**
- The table represents a one-to-one function since each \( x \)-value has a unique corresponding \( f(x) \)-value, and vice versa.

3. **Determine the inverse of the function:**
- To find the inverse of \( f(x) = 2x \), swap \( x \) and \( f(x) \). This gives the equation \( f^{-1}(x) = \frac{x}{2} \).

4. **Create a new table for the inverse:**
- Swap the \( x \)- and \( f(x) \)-values from the original table to create the inverse table:

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

5. **Verify the inverse:**
- Apply the inverse function \( f^{-1}(x) = \frac{x}{2} \) to the values in the inverse table:
- \( f^{-1}(2) = \frac{2}{2} = 1 \)
- \( f^{-1}(4) = \frac{4}{2} = 2 \)
- \( f^{-1}(6) = \frac{6}{2} = 3 \)
- \( f^{-1}(8) = \frac{8}{2} = 4 \)

This confirms that the inverse function is correct.

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### **Final Answer:**

The inverse function of \( f(x) = 2x \) is \( f^{-1}(x) = \frac{x}{2} \), and the table for the inverse is:

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

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Nick Perich
Norristown Area High School
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