Explain how to find the inverse of a quadratic function when graphed and reflect it over the line \( y = x \); provide an example using the function \( h(x) = x^2 - 4 \).

To find the inverse of a quadratic function by reflecting its graph over the line \( y = x \), you need to follow a series of steps. Since quadratic functions are **not one-to-one** over their entire domain (they fail the horizontal line test), you'll first need to **restrict the domain** to make the function one-to-one. Let's walk through this process using the function \( h(x) = x^2 - 4 \) as an example. ### Step 1: Understand the Original Function The function \( h(x) = x^2 - 4 \) is a parabola that opens upwards with its vertex at \( (0, -4) \). Its graph is symmetric about the y-axis. ### Step 2: Restrict the Domain To make \( h(x) \) one-to-one, restrict its domain to either \( x \geq 0 \) or \( x \leq 0 \). This ensures that for each \( y \)-value, there's only one corresponding \( x \)-value. **Example Restriction:** Let's choose \( x \geq 0 \). ### Step 3: Reflect the Graph Over \( y = x \) Reflecting a function over the line \( y = x \) effectively swaps the roles of \( x \) and \( y \). To find the inverse: 1. **Start with the original equation:** \[ y = x^2 - 4 \] 2. **Swap \( x \) and \( y \):** \[ x = y^2 - 4 \] 3. **Solve for \( y \):** \[ x + 4 = y^2 \\ y = \sqrt{x + 4} \] *Note:* Since we restricted the domain to \( x \geq 0 \), we take the **positive** square root. ### Step 4: Define the Inverse Function The inverse function is: \[ h^{-1}(x) = \sqrt{x + 4} \] ### Step 5: Verify the Inverse To ensure that \( h^{-1}(x) \) is indeed the inverse, verify that: \[ h(h^{-1}(x)) = x \quad \text{and} \quad h^{-1}(h(x)) = x \] **Verification:** 1. **First Composition:** \[ h(h^{-1}(x)) = h(\sqrt{x + 4}) = (\sqrt{x + 4})^2 - 4 = x + 4 - 4 = x \] 2. **Second Composition:** \[ h^{-1}(h(x)) = h^{-1}(x^2 - 4) = \sqrt{(x^2 - 4) + 4} = \sqrt{x^2} = x \] *Note:* This holds true because \( x \geq 0 \). ### Step 6: Graphical Interpretation - **Original Function \( h(x) = x^2 - 4 \):** A parabola opening upwards with vertex at \( (0, -4) \) and restricted to \( x \geq 0 \). - **Inverse Function \( h^{-1}(x) = \sqrt{x + 4} \):** This is the right half of a sideways parabola, reflecting the original function over the line \( y = x \). When graphed together, these two functions will be mirror images of each other across the line \( y = x \). ### Complete Example **Original Function:** \[ h(x) = x^2 - 4 \quad \text{for} \quad x \geq 0 \] **Finding the Inverse:** 1. **Swap \( x \) and \( y \):** \[ x = y^2 - 4 \] 2. **Solve for \( y \):** \[ y^2 = x + 4 \\ y = \sqrt{x + 4} \] 3. **Inverse Function:** \[ h^{-1}(x) = \sqrt{x + 4} \] **Verification:** - \( h(h^{-1}(x)) = x \) - \( h^{-1}(h(x)) = x \) (since \( x \geq 0 \)) **Graphical Reflection:** - Plot \( h(x) = x^2 - 4 \) (restricted to \( x \geq 0 \)). - Plot \( h^{-1}(x) = \sqrt{x + 4} \). - These graphs will be symmetrical about the line \( y = x \). ### Important Considerations - **Domain Restriction:** Without restricting the domain, the inverse wouldn't be a function because it wouldn't pass the vertical line test after reflection. Always ensure the original function is one-to-one before finding its inverse. - **Choosing Branches:** Depending on the domain restriction (e.g., \( x \geq 0 \) or \( x \leq 0 \)), the inverse will correspondingly take the positive or negative square root. By following these steps, you can successfully find the inverse of any quadratic function by reflecting its graph over the line \( y = x \), provided you properly restrict its domain to ensure it's one-to-one.