If a given equation has two solutions x = 3 and x = 5 for the same y value, what
can you conclude?
(i) The equation cannot represent a function f for which y = f(x)
(ii) The equation could represent a function f for which y = f(x)

Given the information that an equation has two solutions, \(x = 3\) and \(x = 5\), for the same \(y\) value, we can conclude the following:

### Explanation:

1. **Definition of a Function:**
   A function \(f(x)\) is a relation where each input \(x\) is associated with exactly one output \(y\). This means that for a function, a specific \(x\) value can produce only one corresponding \(y\) value.

2. **Given Condition:**
   The problem states that both \(x = 3\) and \(x = 5\) yield the same \(y\) value. Mathematically, this is represented as:
   \[
   f(3) = y \quad \text{and} \quad f(5) = y
   \]
   
3. **Implication of the Given Condition:**
   - If an equation provides the same \(y\) value for different \(x\) values (like \(x = 3\) and \(x = 5\)), it does not violate the definition of a function. 
   - A function can have different inputs \(x\) that result in the same output \(y\); however, it cannot have a single input \(x\) that results in more than one output \(y\).

### Conclusion:

- **Statement (i) is Incorrect:** The equation *can* represent a function. The condition that multiple \(x\) values map to the same \(y\) does not violate the definition of a function.
- **Statement (ii) is Correct:** The equation could represent a function \(f\) for which \(y = f(x)\). This situation aligns with the properties of functions, where multiple inputs can yield the same output.

Thus, the correct conclusion is:

\[
\text{(ii) The equation could represent a function } f \text{ for which } y = f(x).
\]

Would you like more details or have any questions?

### Related Questions:

1. What are the conditions for an equation to represent a function?
2. Can a function have an infinite number of different inputs producing the same output?
3. How does the vertical line test help determine if a graph represents a function?
4. What is the difference between a function and a relation?
5. How can you determine the domain and range of a function from an equation?

**Tip:** Remember, a function can map multiple inputs to the same output, but each input must have a unique output!

Explore why an equation with multiple solutions, x = 3 and x = 5 for the same y value, can still represent a function. Learn about the definition of a function and when multiple inputs can yield the same output. Suitable for Grades 9-12.

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