The function f is defined as follows.
f left parenthesis x right parenthesis equalsleft brace Start 2 By 1 Matrix 1st Row 1st Column 4 plus x 2nd Row 1st Column x squared EndMatrix Start 2 By 1 Matrix 1st Row 1st Column if x less than 0 2nd Row 1st Column if x greater than or equals 0 EndMatrix
 
 
​(a) Find the domain of the function.
​(b) Locate any intercepts.
​(c) Graph the function.
​(d) Based on the​ graph, find the range.
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Part 1
​(a) The domain of the function f is 
  
enter your response here.
​(Type your answer in interval​ notation.)

### (a) Domain of the function

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

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

The domain of a function consists of all possible values of \( x \) for which the function is defined. Since both pieces of the function \( 4 + x \) and \( x^2 \) are defined for all real values of \( x \), the domain includes all real numbers. 

Thus, the domain is:

\[
\boxed{(-\infty, \infty)}
\]

### (b) Intercepts

#### x-intercept:
- The x-intercept occurs where \( f(x) = 0 \).
  
For \( x < 0 \), \( f(x) = 4 + x \). Set \( 4 + x = 0 \):

\[
4 + x = 0 \implies x = -4
\]

For \( x \geq 0 \), \( f(x) = x^2 \). Set \( x^2 = 0 \):

\[
x = 0
\]

Thus, the x-intercepts are \( x = -4 \) and \( x = 0 \).

#### y-intercept:
- The y-intercept occurs where \( x = 0 \).

At \( x = 0 \), we use the second piece of the function since \( x = 0 \geq 0 \). So, \( f(0) = 0^2 = 0 \).

Therefore, the y-intercept is \( (0, 0) \).

### (c) Graph of the function

The function consists of two parts:
1. For \( x < 0 \), the function is a linear function \( f(x) = 4 + x \).
2. For \( x \geq 0 \), the function is a quadratic function \( f(x) = x^2 \).

At \( x = 0 \), the two pieces of the function meet at \( (0, 0) \), and the graph transitions smoothly from the linear piece to the quadratic piece. I will generate a graph of this function.

### (d) Range of the function

Based on the graph:
- For \( x < 0 \), \( f(x) = 4 + x \), which decreases as \( x \) decreases. The maximum value is \( f(0) = 0 \), and as \( x \to -\infty \), \( f(x) \to -\infty \).
- For \( x \geq 0 \), \( f(x) = x^2 \), which is non-negative and increases as \( x \) increases. The minimum value is \( f(0) = 0 \), and there is no upper bound.

Thus, the range is:

\[
\boxed{(-\infty, \infty)}
\]

---

Would you like me to graph the function for you or need more details?

### Related questions:
1. How do you find the range from the equation alone?
2. What is the significance of the domain in piecewise functions?
3. Can there be more than one y-intercept for piecewise functions?
4. How do quadratic functions affect the shape of graphs?
5. Why does the function's behavior change at \( x = 0 \)?

**Tip:** For piecewise functions, make sure to evaluate the function at the boundary points of each piece to ensure continuity or determine any breaks in the graph.

The function f is defined as follows. f(x) = { 4 + 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.

Explore the domain, intercepts, graph, and range of the piecewise function f(x), which includes a linear part for x < 0 and a quadratic part for x ≥ 0. Suitable for students in Grades 9-11, this problem provides a detailed explanation of how to handle functions defined by multiple rules.

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