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.

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.