Let ` f(x) =| x+1 | +|2^x +1| ,` then f(x) can be rewritten in the form ` f(x) ={{:(2^x -x , : x lt -1),(2^x + x+2 , : x ge -1):}`

Answer

To rewrite the function \( f(x) = |x + 1| + |2^x + 1| \) in the specified piecewise form, we will analyze the function based on the conditions of the absolute values.

### Step 1: Identify the critical points
The absolute value expressions change at points where their arguments equal zero.

1. For \( |x + 1| \):
- Set \( x + 1 = 0 \) → \( x = -1 \)

2. For \( |2^x + 1| \):
- Set \( 2^x + 1 = 0 \) → \( 2^x = -1 \) (This has no solution since \( 2^x \) is always positive.)

Thus, the only critical point is \( x = -1 \).

### Step 2: Define the intervals
We will consider two intervals based on the critical point:
1. \( x < -1 \)
2. \( x \geq -1 \)

### Step 3: Evaluate \( f(x) \) for each interval

#### Case 1: \( x < -1 \)
In this case:
- \( x + 1 < 0 \) → \( |x + 1| = -(x + 1) = -x - 1 \)
- \( 2^x + 1 > 0 \) (since \( 2^x \) is always positive) → \( |2^x + 1| = 2^x + 1 \)

Thus, for \( x < -1 \):
\[ f(x) = -x - 1 + (2^x + 1) = -x + 2^x \]

#### Case 2: \( x \geq -1 \)
In this case:
- \( x + 1 \geq 0 \) → \( |x + 1| = x + 1 \)
- \( 2^x + 1 > 0 \) → \( |2^x + 1| = 2^x + 1 \)

Thus, for \( x \geq -1 \):
\[ f(x) = (x + 1) + (2^x + 1) = x + 2^x + 2 \]

### Step 4: Combine the results into piecewise function
Now we can combine the results from both cases into a piecewise function:
\[ f(x) = \begin{cases} 2^x - x & \text{if } x < -1 \\ 2^x + x + 2 & \text{if } x \geq -1 \end{cases} \]

### Final Answer
Thus, the function can be rewritten as:
\[ f(x) = \begin{cases} 2^x - x & \text{if } x < -1 \\ 2^x + x + 2 & \text{if } x \geq -1 \end{cases} \]