For which of the following, y can be a function of x, `(x in R, y in R)`?
`{:((i) (x-h)^(2)+(y-k)^(2)=r^(2),(ii)y^(2)=4ax),((iii) x^(4)=y^(2),(iv) x^(6)=y^(3)),((v) 3y=(log x)^(2),""):}`

Answer

To determine for which of the given equations \( y \) can be a function of \( x \), we need to analyze each equation based on the definition of a function. A relation \( y \) is a function of \( x \) if for every value of \( x \), there is exactly one corresponding value of \( y \).

Let's analyze each equation step by step:

### Step 1: Analyze the first equation
**Equation:** \( (x-h)^2 + (y-k)^2 = r^2 \)

This equation represents a circle centered at \( (h, k) \) with radius \( r \). For a given \( x \), there can be two corresponding values of \( y \) (one above and one below the center \( k \)).

**Conclusion:** This is **not a function** of \( x \).

### Step 2: Analyze the second equation
**Equation:** \( y^2 = 4ax \)

This is a parabola that opens to the right. For a given positive value of \( x \), there are two corresponding values of \( y \) (one positive and one negative).

**Conclusion:** This is **not a function** of \( x \).

### Step 3: Analyze the third equation
**Equation:** \( x^4 = y^2 \)

This equation can be rewritten as \( y = \pm x^2 \). For each \( x \), there are two corresponding values of \( y \) (one positive and one negative).

**Conclusion:** This is **not a function** of \( x \).

### Step 4: Analyze the fourth equation
**Equation:** \( x^6 = y^3 \)

This can be rewritten as \( y = x^{6/3} = x^2 \) (taking the cube root). Here, for each value of \( x \), there is exactly one corresponding value of \( y \).

**Conclusion:** This **is a function** of \( x \).

### Step 5: Analyze the fifth equation
**Equation:** \( 3y = (\log x)^2 \)

This can be rewritten as \( y = \frac{(\log x)^2}{3} \). For every positive value of \( x \), there is exactly one corresponding value of \( y \).

**Conclusion:** This **is a function** of \( x \).

### Final Summary
The equations for which \( y \) can be a function of \( x \) are:
- (iv) \( x^6 = y^3 \)
- (v) \( 3y = (\log x)^2 \)