Which of the following functions is graphed below? ￼ ￼
A.
𝑦
=
{
𝑥
3
−
4
,
𝑥
<
1
𝑥
2
+
3
,
𝑥
≤
1
y={ 
x 
3
 −4,x<1
x 
2
 +3,x≤1
​
 
￼
B.
𝑦
=
{
𝑥
3
−
4
,
𝑥
≤
1
𝑥
2
+
3
,
𝑥
>
1
y={ 
x 
3
 −4,x≤1
x 
2
 +3,x>1
​￼
C.
𝑦
=
{
𝑥
3
−
4
,
𝑥
<
1
𝑥
2
+
3
,
𝑥
≥
1
y={ 
x 
3
 −4,x<1
x 
2
 +3,x≥1
​
 
￼
D.
𝑦
=
{
𝑥
3
−
4
,
𝑥
>
1
𝑥
2
+
3
,
𝑥
≤
1
y={ 
x 
3
 −4,x>1
x 
2
 +3,x≤1
​

Let's carefully analyze each of the given piecewise functions and the conditions for each piece, and then determine which one best matches the description of the graph.

1. **A**: 
   \[
   y = \begin{cases} 
   x^3 - 4, & \text{if } x < 1 \\
   x^2 + 3, & \text{if } x \leq 1
   \end{cases}
   \]
   This implies that for \(x < 1\), the function follows \(y = x^3 - 4\), and at \(x = 1\), the function switches to \(y = x^2 + 3\). However, there is ambiguity at \(x = 1\), as it belongs to both parts.

2. **B**:
   \[
   y = \begin{cases} 
   x^3 - 4, & \text{if } x \leq 1 \\
   x^2 + 3, & \text{if } x > 1
   \end{cases}
   \]
   In this case, for \(x \leq 1\), the function follows \(y = x^3 - 4\), and for \(x > 1\), it follows \(y = x^2 + 3\). The condition at \(x = 1\) is clearly specified here.

3. **C**:
   \[
   y = \begin{cases} 
   x^3 - 4, & \text{if } x < 1 \\
   x^2 + 3, & \text{if } x \geq 1
   \end{cases}
   \]
   Here, the function uses \(y = x^3 - 4\) for \(x < 1\), and at \(x = 1\), it switches to \(y = x^2 + 3\). 

4. **D**:
   \[
   y = \begin{cases} 
   x^3 - 4, & \text{if } x > 1 \\
   x^2 + 3, & \text{if } x \leq 1
   \end{cases}
   \]
   This suggests that for \(x > 1\), the function follows \(y = x^3 - 4\), which conflicts with the idea that cubic functions should decrease for negative \(x\).

### Key Consideration:
For a proper piecewise function graph, we expect smooth transitions or at least continuity in the function values. The most likely option is **B**, because:
- It clearly defines the cubic \(x^3 - 4\) for \(x \leq 1\) and the quadratic \(x^2 + 3\) for \(x > 1\).
- This also avoids ambiguity at \(x = 1\).

Thus, the correct answer is **B**.

---

Would you like more details on graphing piecewise functions or the transitions between them?

### 5 Related Questions:
1. How do you graph piecewise functions manually?
2. What is the difference between a cubic and a quadratic function?
3. How do you find where a piecewise function is continuous?
4. How can you determine if a piecewise function has a jump discontinuity?
5. What are some real-world applications of piecewise functions?

### Tip:
When graphing piecewise functions, always check the transition points to ensure proper matching or discontinuity between the segments.

Which of the following functions is graphed below?
A. y = {x^3 - 4, x < 1; x^2 + 3, x ≤ 1}
B. y = {x^3 - 4, x ≤ 1; x^2 + 3, x > 1}
C. y = {x^3 - 4, x < 1; x^2 + 3, x ≥ 1}
D. y = {x^3 - 4, x > 1; x^2 + 3, x ≤ 1}

This page provides an in-depth analysis of a piecewise function, featuring cubic and quadratic parts. Learn how to interpret the conditions for each segment and solve for the correct graph. Suitable for Grades 10-12, with key concepts in function continuity and transition analysis.

Math Problem Statement

Which of the following functions is graphed below? A. 𝑦

B. 𝑦

1 y={ x 3 −4,x≤1 x 2 +3,x>1 C. 𝑦

D. 𝑦

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