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To determine if each given equation defines \( y \) as a function of \( x \), let's analyze each part individually. For \( y \) to be a function of \( x \), each \( x \)-value must correspond to exactly one \( y \)-value.

### Part A: \( x - 1 = y^2 \)
1. Rearrange the equation to solve for \( y \):
   \[
   y^2 = x - 1
   \]
2. Taking the square root of both sides, we get:
   \[
   y = \pm \sqrt{x - 1}
   \]
This equation produces two values of \( y \) (positive and negative roots) for each \( x \) where \( x > 1 \). Hence, this does **not** define \( y \) as a function of \( x \) because it fails the vertical line test (some \( x \)-values correspond to two \( y \)-values).

**Answer for A**: **No**, it does not define \( y \) as a function of \( x \).

---

### Part B: Graph Analysis
To determine if the graph in Part B represents \( y \) as a function of \( x \), apply the **vertical line test**: if any vertical line intersects the graph at more than one point, then \( y \) is not a function of \( x \).

In this case, the graph in Part B intersects vertical lines at multiple points in certain regions, meaning there are multiple \( y \)-values for some \( x \)-values.

**Answer for B**: **No**, it does not define \( y \) as a function of \( x \).

---

### Part C: \( 1 + x^2 = y - 3 \)
1. Rearrange to solve for \( y \):
   \[
   y = 1 + x^2 + 3
   \]
   Simplifies to:
   \[
   y = x^2 + 4
   \]
This is a quadratic function in \( x \), and since quadratic functions open upwards (for \( x^2 \) terms) and have only one \( y \)-value for each \( x \)-value, this equation **does** define \( y \) as a function of \( x \).

**Answer for C**: **Yes**, it defines \( y \) as a function of \( x \).

---

Let me know if you'd like further clarification or a detailed explanation of any part. 

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### Follow-up Questions
1. What is the vertical line test and why is it important for functions?
2. How does the square root in Part A affect the function definition?
3. Can you give examples of other equations that do not define \( y \) as a function of \( x \)?
4. How would the answer change if we had \( y = x^2 - 4 \) instead?
5. What is the general form of quadratic functions and how do they relate to function definitions?

### Tip
When checking if an equation defines \( y \) as a function of \( x \), always solve for \( y \) and check if any \( x \)-value leads to multiple \( y \)-values. This ensures clarity on whether it passes the vertical line test.

Determine if the following define y as a function of x: A) x - 1 = y^2, B) (graph shown), C) 1 + x^2 = y - 3.

Analyze if equations such as x - 1 = y^2 and 1 + x^2 = y - 3, as well as a provided graph, define y as a function of x. Learn to use the vertical line test and solve for y in each equation. Suitable for high school students studying algebra and functions.

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