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 is a function, we need to apply the definition of a function. A relation is considered a function if for every input (x-value), there is exactly one output (y-value).

Let’s analyze the options step by step:

### Step 1: Understand the Definition of a Function
A function is defined as a relation where each input is associated with exactly one output. This means that for any value of x, there should be only one corresponding value of y.

### Step 2: Analyze Each Option
We will analyze each of the options provided to see if they meet the criteria of a function.

#### Option 1: \( y^2 = x \)
- For \( x = 8 \), we have \( y^2 = 8 \).
- This gives us \( y = \sqrt{8} \) and \( y = -\sqrt{8} \).
- Since there are two values of y for one value of x, this is **not a function**.

#### Option 2: \( y = \text{greatest integer}(x) \)
- The greatest integer function (floor function) assigns to each x the largest integer less than or equal to x.
- For any value of x, there is exactly one corresponding integer value of y.
- Therefore, this is a **function**.

#### Option 3: \( y^2 = 4ax \)
- This can be rewritten as \( y = 2\sqrt{ax} \) and \( y = -2\sqrt{ax} \).
- For a positive value of x, there are two corresponding values of y (positive and negative).
- Hence, this is **not a function**.

#### Option 4: \( y^2 = x^4 \)
- This gives us \( y = x^2 \) and \( y = -x^2 \).
- For each positive value of x, there are two corresponding values of y (positive and negative).
- Therefore, this is **not a function**.

### Step 3: Conclusion
After analyzing all the options, we find that:
- **Option 2** is the only one that qualifies as a function.

### Final Answer
The correct option that represents a function is **Option 2**.

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