create me a piecewise function so I can solve it

Sure, here's a piecewise function for you to solve:

\[ f(x) = \begin{cases} 
x^2 + 2x + 1 & \text{if } x < 0 \\
3x + 4 & \text{if } 0 \leq x < 5 \\
2x - 1 & \text{if } x \geq 5 
\end{cases} \]

Let's solve for \( f(x) \) at specific values:

1. \( f(-2) \)
2. \( f(0) \)
3. \( f(3) \)
4. \( f(5) \)
5. \( f(7) \)

### Solutions:

1. **\( f(-2) \)**:
   Since \( -2 < 0 \),
   \[ f(-2) = (-2)^2 + 2(-2) + 1 = 4 - 4 + 1 = 1 \]

2. **\( f(0) \)**:
   Since \( 0 \leq x < 5 \),
   \[ f(0) = 3(0) + 4 = 0 + 4 = 4 \]

3. **\( f(3) \)**:
   Since \( 0 \leq x < 5 \),
   \[ f(3) = 3(3) + 4 = 9 + 4 = 13 \]

4. **\( f(5) \)**:
   Since \( x \geq 5 \),
   \[ f(5) = 2(5) - 1 = 10 - 1 = 9 \]

5. **\( f(7) \)**:
   Since \( x \geq 5 \),
   \[ f(7) = 2(7) - 1 = 14 - 1 = 13 \]

Do you want details on any of these calculations, or do you have any questions?

### Related Questions:
1. How do you define a piecewise function?
2. What is the domain of a piecewise function?
3. How do you find the range of a piecewise function?
4. How do you graph a piecewise function?
5. How do you solve for the derivative of a piecewise function?
6. How do you solve for the integral of a piecewise function?
7. How do you determine if a piecewise function is continuous?
8. What are the applications of piecewise functions in real life?

### Tip:
When dealing with piecewise functions, always pay attention to the defined intervals to ensure you're using the correct expression for each segment of the function.

Explore how to solve piecewise function problems step-by-step. Learn about evaluating piecewise functions at specific points and understanding different cases based on the defined intervals. Suitable for high school students studying advanced algebra and functions.

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