If `f(x)=x^(2)+2x+2,x ge-1`
`=x^(3)+3x+1, x lt-1`
then find the value of `f(f(-2)):`

Options

A. 15
B. 17
C. 22
D. none of these

Answer

To find the value of \( f(f(-2)) \) given the piecewise function:

\[ f(x) = \begin{cases} x^2 + 2x + 2 & \text{if } x \geq -1 \\ x^3 + 3x + 1 & \text{if } x < -1 \end{cases} \]

we will follow these steps:

### Step 1: Calculate \( f(-2) \)

Since \(-2 < -1\), we will use the second case of the piecewise function:

\[ f(-2) = (-2)^3 + 3(-2) + 1 \]

Calculating this:

\[ = -8 - 6 + 1 \]
\[ = -14 + 1 \]
\[ = -13 \]

Thus, \( f(-2) = -13 \).

### Step 2: Calculate \( f(f(-2)) = f(-13) \)

Now we need to find \( f(-13) \). Since \(-13 < -1\), we again use the second case of the piecewise function:

\[ f(-13) = (-13)^3 + 3(-13) + 1 \]

Calculating this:

\[ = -2197 - 39 + 1 \]
\[ = -2197 - 39 + 1 \]
\[ = -2236 + 1 \]
\[ = -2235 \]

Thus, \( f(-13) = -2235 \).

### Final Answer

Therefore, the value of \( f(f(-2)) \) is:

\[ \boxed{-2235} \]