If `f:[-4,0]->R` is defined by `f(x) = e^x + sin x,` its even extension to `[-4, 4]` is given by :

Options

A. `-e^(x)-sinx`
B. `e^(-|x|)-sin|x|`
C. `e^(-|x|)+ sin|x|`
D. `-e^(-|x|+sin|x|)`

Answer

To find the even extension of the function \( f(x) = e^x + \sin x \) defined on the interval \( [-4, 0] \) to the interval \( [-4, 4] \), we can follow these steps:

### Step 1: Understand the Definition of Even Extension
An even extension of a function \( f(x) \) defined on an interval \( [a, b] \) is a function \( g(x) \) defined on \( [-b, b] \) such that:
- \( g(x) = f(x) \) for \( x \in [a, b] \)
- \( g(-x) = g(x) \) for all \( x \in [0, b] \)

### Step 2: Define the Function for the Given Interval
Given:
\[ f(x) = e^x + \sin x \quad \text{for } x \in [-4, 0] \]

### Step 3: Find the Even Extension
For \( x \in [0, 4] \), the even extension \( g(x) \) will be defined as:
\[ g(x) = f(-x) \quad \text{for } x \in [0, 4] \]

Calculating \( f(-x) \):
\[ f(-x) = e^{-x} + \sin(-x) \]
Using the property of sine:
\[ \sin(-x) = -\sin(x) \]
Thus:
\[ f(-x) = e^{-x} - \sin(x) \]

### Step 4: Combine the Definitions
Now we can define \( g(x) \) for the entire interval \( [-4, 4] \):
\[ g(x) = \begin{cases} e^x + \sin x & \text{for } x \in [-4, 0] \\ e^{-x} - \sin x & \text{for } x \in [0, 4] \end{cases} \]

### Step 5: Write the Complete Even Extension
Thus, the even extension \( g(x) \) of the function \( f(x) \) is:
\[ g(x) = \begin{cases} e^x + \sin x & \text{for } x \in [-4, 0] \\ e^{-x} - \sin x & \text{for } x \in [0, 4] \end{cases} \]

### Final Answer
The even extension of the function \( f(x) = e^x + \sin x \) to the interval \( [-4, 4] \) is:
\[ g(x) = \begin{cases} e^x + \sin x & \text{for } x \in [-4, 0] \\ e^{-x} - \sin x & \text{for } x \in [0, 4] \end{cases} \]