Express the function `f(x) =2x^(4) - 3x^(2) + 6x+7-4 sin x` as sum of an even and an odd function.

Answer

To express the function \( f(x) = 2x^4 - 3x^2 + 6x + 7 - 4\sin x \) as the sum of an even and an odd function, we will follow these steps:

### Step 1: Identify the even and odd components
Recall the definitions:
- A function \( g(x) \) is even if \( g(-x) = g(x) \).
- A function \( h(x) \) is odd if \( h(-x) = -h(x) \).

### Step 2: Separate the function into even and odd parts
We can separate \( f(x) \) into two parts:
1. An even part \( g(x) \)
2. An odd part \( h(x) \)

Let’s define:
- \( g(x) = 2x^4 - 3x^2 + 7 \) (even part)
- \( h(x) = 6x - 4\sin x \) (odd part)

### Step 3: Verify that \( g(x) \) is even
To check if \( g(x) \) is even, we compute \( g(-x) \):
\[ g(-x) = 2(-x)^4 - 3(-x)^2 + 7 = 2x^4 - 3x^2 + 7 = g(x) \]
Since \( g(-x) = g(x) \), \( g(x) \) is indeed an even function.

### Step 4: Verify that \( h(x) \) is odd
Now, we check if \( h(x) \) is odd by computing \( h(-x) \):
\[ h(-x) = 6(-x) - 4\sin(-x) = -6x + 4\sin x \]
This can be rewritten as:
\[ h(-x) = - (6x - 4\sin x) = -h(x) \]
Since \( h(-x) = -h(x) \), \( h(x) \) is an odd function.

### Step 5: Combine the even and odd parts
Now we can express \( f(x) \) as the sum of the even and odd functions:
\[ f(x) = g(x) + h(x) = (2x^4 - 3x^2 + 7) + (6x - 4\sin x) \]

### Final Result
Thus, we have expressed the function \( f(x) \) as:
\[ f(x) = g(x) + h(x) = (2x^4 - 3x^2 + 7) + (6x - 4\sin x) \]
where \( g(x) = 2x^4 - 3x^2 + 7 \) is the even function and \( h(x) = 6x - 4\sin x \) is the odd function.

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