A function `f : R rarr` defined by `f(x) = x^(2)`. Determine
`{y : f(y) = - 1}`

Answer

To solve the problem, we need to determine the set of values \( y \) such that \( f(y) = -1 \), where the function \( f \) is defined as \( f(x) = x^2 \).

### Step-by-Step Solution:

1. **Understand the Function**:
The function is given as \( f(x) = x^2 \). This means for any real number \( x \), \( f(x) \) will return the square of \( x \).

2. **Set Up the Equation**:
We need to find \( y \) such that:
\[ f(y) = -1 \]
Substituting the function into the equation gives:
\[ y^2 = -1 \]

3. **Analyze the Equation**:
The equation \( y^2 = -1 \) implies that we are looking for a real number \( y \) whose square equals \(-1\).

4. **Consider the Properties of Squares**:
We know that the square of any real number is always non-negative. Therefore, \( y^2 \) can never be less than 0. This means:
\[ y^2 \geq 0 \quad \text{for all } y \in \mathbb{R} \]
Since \(-1\) is less than \(0\), there are no real numbers \( y \) such that \( y^2 = -1 \).

5. **Conclusion**:
Since there are no real solutions to the equation \( y^2 = -1 \), we conclude that the set of \( y \) such that \( f(y) = -1 \) is the empty set. Therefore, we can write:
\[ \{ y : f(y) = -1 \} = \emptyset \]

### Final Answer:
\[ \{ y : f(y) = -1 \} = \emptyset \]