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Let's solve the questions about the given function \( f(x) \).

The function is defined piecewise as:
\[
f(x) = \begin{cases}
3 + x, & \text{if } x < 0 \\
x^2, & \text{if } x \geq 0
\end{cases}
\]

### (a) Find the domain of the function.
The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case:
- For \( x < 0 \), the function is \( 3 + x \), which is defined for all negative \( x \).
- For \( x \geq 0 \), the function is \( x^2 \), which is defined for all non-negative \( x \).

Thus, the domain of \( f(x) \) is all real numbers \( (-\infty, \infty) \).

### (b) Locate any intercepts.
- **x-intercepts**: These occur where \( f(x) = 0 \).
    - For \( x < 0 \), solve \( 3 + x = 0 \) ⇒ \( x = -3 \).
    - For \( x \geq 0 \), solve \( x^2 = 0 \) ⇒ \( x = 0 \).
    - So, the x-intercepts are at \( x = -3 \) and \( x = 0 \).
- **y-intercept**: This occurs where \( x = 0 \). At \( x = 0 \), \( f(0) = 0^2 = 0 \). So, the y-intercept is at \( (0, 0) \).

### (c) Graph the function.
- For \( x < 0 \), the graph is the line \( f(x) = 3 + x \), which has a slope of 1 and a y-intercept at \( (0, 3) \), but this part only applies for \( x < 0 \).
- For \( x \geq 0 \), the graph is the parabola \( f(x) = x^2 \), starting at \( (0, 0) \).

The graph consists of two parts:
1. A line with slope 1 for \( x < 0 \).
2. A parabola starting at \( (0, 0) \) for \( x \geq 0 \).

### (d) Based on the graph, find the range.
- For \( x < 0 \), \( f(x) = 3 + x \), so the values of \( f(x) \) range from \( (-\infty, 3) \).
- For \( x \geq 0 \), \( f(x) = x^2 \), so the values of \( f(x) \) range from \( [0, \infty) \).

Therefore, combining both parts, the range of the function is \( (-\infty, 3) \cup [0, \infty) \).

Would you like further details on any specific part of this solution?

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Here are 5 related questions to expand on this topic:
1. How do we determine whether a function is continuous or not?
2. What is the difference between the domain and the range of a function?
3. How do piecewise functions apply in real-world problems?
4. Can the function have a different slope in other intervals?
5. How do you calculate the derivative of a piecewise function?

**Tip**: When graphing piecewise functions, always pay attention to whether the conditions include the boundary point, such as using \( \geq \) or \( > \) at key points.

The function f is defined as f(x) = {3 + x if x < 0, x^2 if x ≥ 0}. (a) Find the domain of the function. (b) Locate any intercepts. (c) Graph the function. (d) Based on the graph, find the range.

Solve a piecewise function defined by f(x) = 3 + x for x < 0 and f(x) = x^2 for x ≥ 0. Find the domain, intercepts, and graph the function. Suitable for Grades 9-12.

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