Which one of the following is a function

Options

A. `y=[x]`
B. `y^(2)=4ax`
C. `x^(4)=y^(2)`
D. `3y=(logx)^(2)`

Answer

To determine which of the given options represents a function, we need to analyze each option based on the definition of a function. A function is a relation where each input (x-value) corresponds to exactly one output (y-value).

### Step-by-Step Solution:

1. **Option 1: \( y = \lfloor x \rfloor \) (Greatest Integer Function)**
- The greatest integer function (floor function) returns the largest integer less than or equal to \( x \).
- For any value of \( x \) in the range [0, 1), \( y \) will be 0.
- For any value of \( x \) in the range [1, 2), \( y \) will be 1.
- Since for each input \( x \), there is only one corresponding output \( y \), this is a function.

**Conclusion**: This option is a function.

2. **Option 2: \( y^2 = 4ax \)**
- Rearranging gives \( y = \pm 2\sqrt{ax} \).
- For a specific value of \( x \) (e.g., \( x = 4 \)), \( y \) can be \( +4\sqrt{a} \) or \( -4\sqrt{a} \).
- Since one input \( x \) can produce two different outputs \( y \), this is not a function.

**Conclusion**: This option is not a function.

3. **Option 3: \( y^2 = x^4 \)**
- Rearranging gives \( y = \pm x^2 \).
- For a specific value of \( x \) (e.g., \( x = 2 \)), \( y \) can be \( +4 \) or \( -4 \).
- Again, one input \( x \) can produce two different outputs \( y \), so this is not a function.

**Conclusion**: This option is not a function.

4. **Option 4: \( 3y = (\log x)^2 \)**
- Rearranging gives \( y = \frac{(\log x)^2}{3} \).
- For each positive value of \( x \), \( \log x \) produces a unique value, and squaring it will also yield a unique result.
- Therefore, for each input \( x \), there is exactly one output \( y \).

**Conclusion**: This option is a function.

### Final Answer:
- The options that represent functions are **Option 1** and **Option 4**.