1. **Problem statement:** Calculate the value of the piecewise function
$$Z = \begin{cases} \sqrt{x^2 + 2x - 1 - 3y + 2y^3}, & x > y \\ |2x^3 + 4y^2| - 3x^4, & x < y \\ \sqrt[3]{4y^3 + 2x^2}, & x = y \end{cases}$$
where $x,y \in \mathbb{R}$.
2. **Formula and rules:**
- For $x > y$, compute the square root of the expression inside, ensuring the radicand is non-negative.
- For $x < y$, compute the absolute value of $2x^3 + 4y^2$ minus $3x^4$.
- For $x = y$, compute the cube root of $4y^3 + 2x^2$.
3. **Step-by-step evaluation:**
- Check the relation between $x$ and $y$.
- If $x > y$:
- Calculate $r = x^2 + 2x - 1 - 3y + 2y^3$.
- Verify $r \geq 0$ to ensure the square root is defined.
- Compute $Z = \sqrt{r}$.
- If $x < y$:
- Calculate $a = 2x^3 + 4y^2$.
- Compute $Z = |a| - 3x^4$.
- If $x = y$:
- Calculate $c = 4y^3 + 2x^2$.
- Compute $Z = \sqrt[3]{c}$.
4. **Python program to compute $Z$ given $x$ and $y$:**
```python
import math
def compute_Z(x, y):
if x > y:
r = x**2 + 2*x - 1 - 3*y + 2*y**3
if r < 0:
raise ValueError('Expression under square root is negative')
Z = math.sqrt(r)
elif x < y:
a = 2*x**3 + 4*y**2
Z = abs(a) - 3*x**4
else: # x == y
c = 4*y**3 + 2*x**2
# Cube root can be computed as pow with exponent 1/3, handling negative values
if c >= 0:
Z = c**(1/3)
else:
Z = -(-c)**(1/3)
return Z
# Example usage:
# x_val, y_val = 2, 1
# print(compute_Z(x_val, y_val))
```
This program checks the condition between $x$ and $y$, computes the corresponding expression, and returns the value of $Z$.
**Final answer:** The function $Z$ is computed as above depending on the relation between $x$ and $y$.
Piecewise Function Z2
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