Determine  Whether the function of the equation defines why y as  a function of x.  X^2 +(y-2)^2=5

To determine whether the equation \( x^2 + (y - 2)^2 = 5 \) defines \( y \) as a function of \( x \), we need to check if each value of \( x \) corresponds to exactly one value of \( y \).

### Step-by-Step Analysis:

1. **Rewrite the equation:**  
   The given equation is:
   \[
   x^2 + (y - 2)^2 = 5
   \]
   This equation represents a circle with the center at \( (0, 2) \) and radius \( \sqrt{5} \).

2. **Determine if \( y \) is a function of \( x \):**  
   For \( y \) to be a function of \( x \), each value of \( x \) should correspond to only one value of \( y \). However, in a circle, a single value of \( x \) can correspond to two different values of \( y \).

   For example, consider the equation when \( x = 0 \):
   \[
   0^2 + (y - 2)^2 = 5 \implies (y - 2)^2 = 5.
   \]
   Solving for \( y \):
   \[
   y - 2 = \pm \sqrt{5}, \quad \text{so} \quad y = 2 + \sqrt{5} \quad \text{or} \quad y = 2 - \sqrt{5}.
   \]
   Here, for \( x = 0 \), there are two possible values for \( y \).

3. **Conclusion:**  
   Since one value of \( x \) (like \( x = 0 \)) corresponds to two different values of \( y \), the equation does **not** define \( y \) as a function of \( x \).

### Additional Explanation:
- A function \( y = f(x) \) must pass the **vertical line test**: A vertical line drawn at any \( x \)-value on its graph must intersect the graph at most once. The equation \( x^2 + (y - 2)^2 = 5 \) represents a circle, which fails this test.

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### Related Questions:
1. What is the domain and range of a circle defined by \( x^2 + (y - 2)^2 = 5 \)?
2. How can we graph the equation \( x^2 + (y - 2)^2 = 5 \) and visualize why it isn't a function?
3. What is the vertical line test, and how does it determine if an equation is a function?
4. How do you solve for \( y \) explicitly in the equation \( x^2 + (y - 2)^2 = 5 \)?
5. What is the difference between a function and a relation in mathematics?

**Tip:** Remember that for a relation to be a function, every input (x-value) must correspond to exactly one output (y-value).

Determine whether the equation defines y as a function of x. x^2 + (y-2)^2 = 5

Analyze the equation x^2 + (y - 2)^2 = 5 to determine if y is a function of x. Explore the concept of the vertical line test and understand why a circle fails to define y as a function of x. Suitable for high school students, Grades 9-12.

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