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The given function \( f(x) \) is piecewise defined as:

\[
f(x) = 
\begin{cases} 
4x^2 + 5 & \text{if } x < 0 \\
5 - 2x^2 & \text{if } x \geq 0
\end{cases}
\]

We are asked to find \( f^{-1}(y) \) for \( y > 5 \).

### 1. **Step-by-Step Analysis**

For \( y > 5 \), we need to consider which part of the piecewise function gives values greater than 5.

#### Case 1: \( f(x) = 4x^2 + 5 \) when \( x < 0 \)
- The function is always positive and increasing for \( x < 0 \), and since \( 4x^2 + 5 \geq 5 \), it cannot give values greater than 5. Therefore, this case does not provide \( y > 5 \).

#### Case 2: \( f(x) = 5 - 2x^2 \) when \( x \geq 0 \)
- For \( 5 - 2x^2 \), it is clear that the function can take values greater than 5 if we reverse the function. To find \( f^{-1}(y) \), we need to solve:

\[
y = 5 - 2x^2
\]

#### Solving for \( x \):

1. Subtract 5 from both sides:
   \[
   y - 5 = -2x^2
   \]

2. Divide by \(-2\):
   \[
   x^2 = \frac{5 - y}{2}
   \]

3. Take the square root:
   \[
   x = \pm \sqrt{\frac{5 - y}{2}}
   \]

Since \( x \geq 0 \), we only take the positive root:
\[
x = \sqrt{\frac{5 - y}{2}}
\]

Thus, \( f^{-1}(y) = \sqrt{\frac{5 - y}{2}} \) for \( y > 5 \).

### Final Answer:
\( f^{-1}(y) = \sqrt{\frac{5 - y}{2}} \) for \( y > 5 \).

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Would you like more details or have any questions?

Here are 5 related questions:
1. How can we compute the inverse of a more complex piecewise function?
2. What conditions must a function meet to have an inverse?
3. How can we determine the domain and range of the inverse function?
4. What happens if \( y = 5 \) in this scenario?
5. How would the solution change if \( y \leq 5 \)?

**Tip**: When finding the inverse of a piecewise function, ensure you carefully consider the range and domain of each piece separately.

Se f(x) = {4x^2 + 5 para x < 0; 5 - 2x^2 para x ≥ 0}, então f⁻¹(y) para y > 5 é igual a:

Learn how to find the inverse function of a piecewise function f(x), where y > 5. This problem involves analyzing quadratic equations and properties of inverse functions, suitable for Grades 9-12.

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