If `f(x)={(x^(2)",","for "x ge0),(x",","for "x lt 0):}`, then fof(x) is given by

Options

A. `x^(2) " for " x ge 0, x " for " x lt 0`
B. `x^(4) " for " x ge 0, x^(2) " for " x lt 0`
C. `x^(4) " for " x ge 0, -x^(2) " for " x lt 0`
D. `x^(4) " for " x ge 0, x " for " x lt 0`

Explanation

To find \( f(f(x)) \) for the given piecewise function

\[ f(x) = \begin{cases} x^2 & \text{for } x \geq 0 \\ x & \text{for } x < 0 \end{cases} \]

we will evaluate \( f(f(x)) \) based on the two cases defined in the function.

### Step 1: Determine \( f(x) \)

1. **Case 1**: If \( x \geq 0 \), then \( f(x) = x^2 \).
2. **Case 2**: If \( x < 0 \), then \( f(x) = x \).

### Step 2: Evaluate \( f(f(x)) \)

Now we will evaluate \( f(f(x)) \) based on the results from Step 1.

#### Sub-case 1: When \( x \geq 0 \)

- Here, \( f(x) = x^2 \).
- Now we need to evaluate \( f(f(x)) = f(x^2) \).

- Since \( x^2 \geq 0 \) (because \( x \geq 0 \)), we use the first case of \( f \):

\[ f(x^2) = (x^2)^2 = x^4 \]

Thus, for \( x \geq 0 \), we have:

\[ f(f(x)) = x^4 \]

#### Sub-case 2: When \( x < 0 \)

- Here, \( f(x) = x \).
- Now we need to evaluate \( f(f(x)) = f(x) \).

- Since \( x < 0 \), we use the second case of \( f \):

\[ f(x) = x \]

Thus, for \( x < 0 \), we have:

\[ f(f(x)) = x \]

### Final Result

Combining both cases, we can write:

\[ f(f(x)) = \begin{cases} x^4 & \text{for } x \geq 0 \\ x & \text{for } x < 0 \end{cases} \]