A function f is defined by ` f(x^(2) ) = x^(3) AA x gt 0 ` then f(4) equals

Options

A. (a)1
B. (b)2
C. (c)8
D. (d) Not differentiable

Answer

To solve the problem, we need to find the value of \( f(4) \) given the function defined by \( f(x^2) = x^3 \) for \( x > 0 \).

### Step-by-Step Solution:

1. **Understand the function definition**: We have \( f(x^2) = x^3 \). This means that for any positive \( x \), if we take the square of \( x \), the function \( f \) will give us the cube of \( x \).

2. **Set up the equation**: We need to find \( f(4) \). To do this, we first need to express \( 4 \) in the form of \( x^2 \). We can set \( x^2 = 4 \).

3. **Solve for \( x \)**: From the equation \( x^2 = 4 \), we take the square root of both sides:
\[ x = \sqrt{4} = 2 \]
Since \( x \) must be greater than 0, we take \( x = 2 \).

4. **Substitute \( x \) back into the function**: Now that we have \( x = 2 \), we can find \( f(4) \) using the function definition:
\[ f(4) = f(x^2) = f(2^2) = 2^3 \]

5. **Calculate \( 2^3 \)**:
\[ 2^3 = 8 \]

6. **Final result**: Therefore, \( f(4) = 8 \).

### Conclusion:
The value of \( f(4) \) is \( 8 \).