The function f is defined as follows.
f left parenthesis x right parenthesis equalsleft brace Start 2 By 1 Matrix 1st Row 1st Column 3 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.

We are given a piecewise function \( f(x) \), defined as:

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

Let's go through each part of the question.

### (a) 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.

- For \( x < 0 \), \( f(x) = 3 + x \), which is defined for all negative values of \( x \).
- For \( x \geq 0 \), \( f(x) = x^2 \), which is defined for all non-negative values of \( x \).

Since there are no restrictions on \( x \) in either part of the piecewise function, the domain of \( f(x) \) is all real numbers.

**Domain:** \( (-\infty, \infty) \)

### (b) Intercepts
**Intercepts** are where the function crosses the x-axis (x-intercepts) or the y-axis (y-intercept).

1. **Y-Intercept:**
   The y-intercept occurs when \( x = 0 \). 
   - Since \( f(x) = x^2 \) when \( x \geq 0 \), we evaluate:
   \[
   f(0) = 0^2 = 0.
   \]
   So, the y-intercept is \( (0, 0) \).

2. **X-Intercepts:**
   X-intercepts occur when \( f(x) = 0 \).

   - For \( x < 0 \), \( f(x) = 3 + x \). Set \( 3 + x = 0 \):
     \[
     x = -3.
     \]
     So, there is an x-intercept at \( (-3, 0) \).
   
   - For \( x \geq 0 \), \( f(x) = x^2 \). Set \( x^2 = 0 \):
     \[
     x = 0.
     \]
     So, there is another x-intercept at \( (0, 0) \), but this is already the y-intercept.

**Intercepts:** \( (-3, 0) \) and \( (0, 0) \)

### (c) Graph of the Function
To graph the function, we consider the behavior of the two pieces:

- For \( x < 0 \), the function is a linear equation \( f(x) = 3 + x \), with slope 1 and y-intercept 3.
- For \( x \geq 0 \), the function is a parabola \( f(x) = x^2 \), which opens upwards with its vertex at \( (0, 0) \).

The two pieces meet at \( x = 0 \), where \( f(0) = 0 \).

We can sketch the graph by plotting key points:
- For \( x < 0 \), plot points like \( (-1, 2) \), \( (-2, 1) \), and \( (-3, 0) \).
- For \( x \geq 0 \), plot points like \( (1, 1) \), \( (2, 4) \), and so on.

Let me know if you want me to plot this for you.

### (d) Range of the Function
The **range** of the function is the set of all possible output values (y-values) the function can take.

- For \( x < 0 \), \( f(x) = 3 + x \), which is a linear function with values approaching 3 as \( x \to 0 \) from the left, and continuing downward as \( x \to -\infty \).
- For \( x \geq 0 \), \( f(x) = x^2 \), which is a parabola that starts at \( f(0) = 0 \) and increases to infinity as \( x \to \infty \).

So, the function covers all values from \( -\infty \) to 3 in the negative region, and all values from 0 to \( \infty \) in the non-negative region. The overall range of the function is:

**Range:** \( (-\infty, \infty) \)

---

Would you like more details on any part, or a graph? Here are some related questions:

1. What would happen to the intercepts if the function were modified?
2. How would the graph change if \( f(x) = x^3 \) for \( x \geq 0 \)?
3. Can the range change for a similar piecewise function if one piece is exponential?
4. How do the continuity and differentiability of this function behave at \( x = 0 \)?
5. What is the slope of the linear part of \( f(x) \) for \( x < 0 \)?

**Tip:** The continuity of a piecewise function is essential to consider at the points where the definition changes.

The function f is defined as follows.
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.

Learn how to find the domain, range, and intercepts of a piecewise function defined as f(x) = 3 + x for x < 0 and f(x) = x^2 for x >= 0. Understand how to graph the function and analyze its key characteristics, suitable for high school students (Grades 9-12).

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