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The function \( f(x) \) is defined as follows:

\[
f(x) = \begin{cases} 
1 - x^2 & \text{if } x \leq 0 \\
2x + 1 & \text{if } x > 0 
\end{cases}
\]

### **Domain:**
The domain of a function is the set of all possible input values (x-values) for which the function is defined.

- For the first piece \( 1 - x^2 \), the function is defined for \( x \leq 0 \).
- For the second piece \( 2x + 1 \), the function is defined for \( x > 0 \).

Combining these, the domain of \( f(x) \) is all real numbers:

\[
\text{Domain of } f(x): \ (-\infty, \infty)
\]

### **Range:**
The range of a function is the set of all possible output values (y-values) that the function can produce.

- For the first piece \( 1 - x^2 \), as \( x \) takes values less than or equal to 0, the maximum value of \( f(x) \) occurs at \( x = 0 \), giving \( f(0) = 1 - 0^2 = 1 \). As \( x \) decreases, \( f(x) \) decreases without bound, so the range is \( (-\infty, 1] \).
  
- For the second piece \( 2x + 1 \), since \( x > 0 \), \( f(x) \) increases without bound as \( x \) increases. The minimum value occurs as \( x \) approaches 0 from the right, giving \( f(x) \) values just slightly greater than 1. Thus, the range is \( (1, \infty) \).

Combining these, the range of \( f(x) \) is:

\[
\text{Range of } f(x): \ (-\infty, \infty)
\]

### Summary:
- **Domain:** \( (-\infty, \infty) \)
- **Range:** \( (-\infty, \infty) \)

Do you want details or have any questions?

Here are 5 related questions to further explore the topic:
1. How would the domain and range change if the condition on \( x \) were modified to \( x < 0 \) and \( x \geq 0 \)?
2. What is the graph of \( f(x) \) based on the given piecewise function?
3. Can you find the points of discontinuity for \( f(x) \)?
4. How would you determine if \( f(x) \) is a continuous function?
5. How do the different parts of the piecewise function interact at the boundary \( x = 0 \)?

**Tip:** When determining the range of a piecewise function, always consider the maximum and minimum values that each piece can achieve within its specified interval.

Explore the domain and range of the piecewise function f(x) = {1 - x^2 if x <= 0, 2x + 1 if x > 0}. Understand how to determine the domain and range for piecewise functions with detailed explanations and examples. Suitable for Grades 10-12 students studying advanced functions.

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