The function
`{:(f:R to R:f(x) = 1, if x gt 0),(" "= 0, if x =0),(" "=-1, if x lt0is a):}`

Options

A. rational function
B. modulus function
C. signum function
D. sinx function

Answer

To solve the given problem, we need to analyze the function defined as follows:

1. **Function Definition**:
- \( f(x) = 1 \) if \( x > 0 \)
- \( f(x) = 0 \) if \( x = 0 \)
- \( f(x) = -1 \) if \( x < 0 \)

2. **Understanding the Function**:
- The function takes different values based on the sign of \( x \).
- For positive values of \( x \), the function outputs \( 1 \).
- For \( x \) equal to \( 0 \), the function outputs \( 0 \).
- For negative values of \( x \), the function outputs \( -1 \).

3. **Graphing the Function**:
- We can plot the function on a Cartesian plane.
- For \( x > 0 \), we draw a horizontal line at \( y = 1 \) (open circle at \( (0, 1) \)).
- At \( x = 0 \), we plot a point at \( (0, 0) \).
- For \( x < 0 \), we draw a horizontal line at \( y = -1 \) (open circle at \( (0, -1) \)).

4. **Identifying the Function Type**:
- The behavior of this function is characteristic of the signum function, often denoted as \( \text{sgn}(x) \).
- The signum function is defined as:
\[ \text{sgn}(x) = \begin{cases} 1 & \text{if } x > 0 \\ 0 & \text{if } x = 0 \\ -1 & \text{if } x < 0 \end{cases} \]

5. **Conclusion**:
- Therefore, the function \( f(x) \) is the signum function.

### Final Answer:
The function \( f(x) \) is the signum function, denoted as \( \text{sgn}(x) \).

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