AP Precalculus Practice Test: Unit 2 Question #19 Find the Inverse of a Quadratic Function

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, Question #19
**Topic:** Finding the Inverse of a Quadratic Function

In this question, you are asked to find the **inverse** of a **quadratic function**. To find the inverse, we must follow certain steps that involve switching the roles of \(x\) and \(y\), and solving for \(y\).

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### **Steps to Find the Inverse of a Quadratic Function:**

1. **Start with the given quadratic function:**
- Let the quadratic function be \( f(x) = ax^2 + bx + c \).

2. **Replace \( f(x) \) with \( y \):**
- Write the function as \( y = ax^2 + bx + c \).

3. **Switch \( x \) and \( y \):**
- Swap \( x \) and \( y \) in the equation to reflect the inverse relationship. This gives:
\[
x = ay^2 + by + c
\]

4. **Solve for \( y \):**
- Solve the equation for \( y \). Since this is a quadratic equation, you will likely need to use the **quadratic formula** or complete the square to isolate \( y \).
- In some cases, depending on the given function, there may be restrictions on the domain to ensure the inverse is a function (since quadratic functions are not one-to-one).

5. **Write the inverse function:**
- Once you solve for \( y \), express it as \( y = f^{-1}(x) \).

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### **Example Problem:**

Given the quadratic function \( f(x) = x^2 + 4x + 3 \), find its inverse.

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### **Solution Steps:**

1. **Start with the given quadratic function:**
\[
f(x) = x^2 + 4x + 3
\]
Replace \( f(x) \) with \( y \):
\[
y = x^2 + 4x + 3
\]

2. **Switch \( x \) and \( y \):**
\[
x = y^2 + 4y + 3
\]

3. **Solve for \( y \):**
- First, subtract 3 from both sides:
\[
x - 3 = y^2 + 4y
\]
- Now, complete the square on the right-hand side. To complete the square for \( y^2 + 4y \), take half of 4 (which is 2), square it (which is 4), and add 4 to both sides:
\[
x - 3 + 4 = y^2 + 4y + 4
\]
\[
x + 1 = (y + 2)^2
\]
- Now, take the square root of both sides:
\[
\sqrt{x + 1} = y + 2
\]
- Finally, subtract 2 from both sides:
\[
y = \sqrt{x + 1} - 2
\]

4. **Write the inverse function:**
The inverse function is:
\[
f^{-1}(x) = \sqrt{x + 1} - 2
\]

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### **Final Answer:**

The inverse of the function \( f(x) = x^2 + 4x + 3 \) is:
\[
f^{-1}(x) = \sqrt{x + 1} - 2
\]

Note that the inverse function has a restricted domain due to the square root, so \( x \geq -1 \).

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

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