Determine whether the equation defines y as a function of x.
a) x^2 + y = 25 b) x + y^2 = 25 c) x = y^3 . show me how to determine 
if they are a function without a calculator


To determine whether an equation defines \( y \) as a function of \( x \), we need to check if, for each value of \( x \), there is only one corresponding value of \( y \). In other words, \( y \) must pass the **vertical line test**, meaning no vertical line should intersect the graph of the equation more than once at any \( x \)-value. Let's examine each case:

### (a) \( x^2 + y = 25 \)

1. **Solve for \( y \):**

   \[
   y = 25 - x^2
   \]

   This is a clear equation for \( y \), and for any value of \( x \), there is only one corresponding value of \( y \). Thus, this equation **defines \( y \) as a function of \( x \)**.

### (b) \( x + y^2 = 25 \)

1. **Solve for \( y \):**

   \[
   y^2 = 25 - x
   \]
   \[
   y = \pm \sqrt{25 - x}
   \]

   This shows that for each value of \( x \), there are two possible values of \( y \) (positive and negative square roots). Therefore, the equation **does not define \( y \) as a function of \( x \)** because a vertical line would intersect the graph at two points for certain \( x \)-values.

### (c) \( x = y^3 \)

1. **Solve for \( y \):**

   \[
   y = \sqrt[3]{x}
   \]

   This equation gives a unique value of \( y \) for each value of \( x \). Since the cube root function is defined for all real numbers and is one-to-one, this equation **defines \( y \) as a function of \( x \)**.

### Summary:
- **(a)** \( x^2 + y = 25 \) → **Yes, function**.
- **(b)** \( x + y^2 = 25 \) → **No, not a function**.
- **(c)** \( x = y^3 \) → **Yes, function**.

Would you like further details on any of these?  
Here are 5 related questions to explore:

1. How do you apply the vertical line test on a graph?
2. Why does solving for \( y \) give multiple values in certain cases?
3. Can we use the horizontal line test to analyze one-to-one functions?
4. How do you determine if an equation involving absolute values defines a function?
5. What is the difference between one-to-one and onto functions?

**Tip:** A key to identifying functions is remembering that each input (x) should correspond to only one output (y).

Determine whether the equation defines y as a function of x. a) x^2 + y = 25 b) x + y^2 = 25 c) x = y^3. Show me how to determine if they are a function without a calculator.

Learn how to determine if y is a function of x for the equations x^2 + y = 25, x + y^2 = 25, and x = y^3. This problem covers key algebraic concepts like the vertical line test and solving for y. Suitable for students in Grades 9-12.

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