If `f(x+2)=x^(2)-x+9`, then what is the value of f(-6)?

Options

A. `-47`
B. `29`
C. `65`
D. `81`

Explanation

To solve the problem, we need to find the value of \( f(-6) \) given that \( f(x + 2) = x^2 - x + 9 \).

### Step 1: Rewrite the function in terms of \( f(x) \)
Since we have \( f(x + 2) \), we can express \( f(x) \) in terms of \( x \) by substituting \( x + 2 \) with a new variable. Let \( y = x + 2 \). Then, we can express \( x \) as \( x = y - 2 \).

### Step 2: Substitute \( x \) in the function
Now, substitute \( x = y - 2 \) into the function:
\[ f(y) = (y - 2)^2 - (y - 2) + 9 \]

### Step 3: Expand the expression
Now, we will expand the expression:
\[ f(y) = (y^2 - 4y + 4) - (y - 2) + 9 \]
\[ = y^2 - 4y + 4 - y + 2 + 9 \]
\[ = y^2 - 5y + 15 \]

### Step 4: Find \( f(-6) \)
Now, we need to find \( f(-6) \):
\[ f(-6) = (-6)^2 - 5(-6) + 15 \]
\[ = 36 + 30 + 15 \]
\[ = 81 \]

### Final Answer
Thus, the value of \( f(-6) \) is \( 81 \).

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