If `f(x)=9x-x^(2)`, `x in R`, then `f(a+1)-f(a-1)` is equal to

Options

A. `4(4-a)`
B. `2(9-2a)`
C. `4(2+a)`
D. `2(4+a)`

Answer

To solve the problem, we need to find the value of \( f(a+1) - f(a-1) \) given the function \( f(x) = 9x - x^2 \).

### Step-by-step Solution:

1. **Define the function**:
We have \( f(x) = 9x - x^2 \).

2. **Calculate \( f(a+1) \)**:
Substitute \( x = a + 1 \) into the function:
\[ f(a+1) = 9(a + 1) - (a + 1)^2 \]
Expanding this:
\[ f(a+1) = 9a + 9 - (a^2 + 2a + 1) \]
Simplifying:
\[ f(a+1) = 9a + 9 - a^2 - 2a - 1 = -a^2 + 7a + 8 \]

3. **Calculate \( f(a-1) \)**:
Substitute \( x = a - 1 \) into the function:
\[ f(a-1) = 9(a - 1) - (a - 1)^2 \]
Expanding this:
\[ f(a-1) = 9a - 9 - (a^2 - 2a + 1) \]
Simplifying:
\[ f(a-1) = 9a - 9 - a^2 + 2a - 1 = -a^2 + 11a - 10 \]

4. **Find \( f(a+1) - f(a-1) \)**:
Now, we subtract \( f(a-1) \) from \( f(a+1) \):
\[ f(a+1) - f(a-1) = (-a^2 + 7a + 8) - (-a^2 + 11a - 10) \]
Simplifying this:
\[ f(a+1) - f(a-1) = -a^2 + 7a + 8 + a^2 - 11a + 10 \]
The \( -a^2 \) and \( +a^2 \) cancel out:
\[ = (7a - 11a) + (8 + 10) = -4a + 18 \]

5. **Factor the result**:
We can factor out a 2:
\[ -4a + 18 = 2(9 - 2a) \]

### Final Answer:
Thus, \( f(a+1) - f(a-1) = 2(9 - 2a) \).