Question

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Explanation:

To solve these problems, we use the function \( f(x) = -x - 11 \) and substitute the given values of \( x \) into the function.

1) \( 4f(-2) + 5f(1) \)

Step 1: Find \( f(-2) \)

Substitute \( x = -2 \) into \( f(x) = -x - 11 \):
\( f(-2) = -(-2) - 11 = 2 - 11 = -9 \)

Step 2: Find \( f(1) \)

Substitute \( x = 1 \) into \( f(x) = -x - 11 \):
\( f(1) = -1 - 11 = -12 \)

Step 3: Calculate \( 4f(-2) + 5f(1) \)

Substitute \( f(-2) = -9 \) and \( f(1) = -12 \):
\( 4(-9) + 5(-12) = -36 - 60 = -96 \)

2) \( 3f(5) \times f(2) \)

Step 1: Find \( f(5) \)

Substitute \( x = 5 \) into \( f(x) = -x - 11 \):
\( f(5) = -5 - 11 = -16 \)

Step 2: Find \( f(2) \)

Substitute \( x = 2 \) into \( f(x) = -x - 11 \):
\( f(2) = -2 - 11 = -13 \)

Step 3: Calculate \( 3f(5) \times f(2) \)

Substitute \( f(5) = -16 \) and \( f(2) = -13 \):
\( 3(-16) \times (-13) = -48 \times (-13) = 624 \)

3) \( -5f(12) - 2f(-9) \)

Step 1: Find \( f(12) \)

Substitute \( x = 12 \) into \( f(x) = -x - 11 \):
\( f(12) = -12 - 11 = -23 \)

Step 2: Find \( f(-9) \)

Substitute \( x = -9 \) into \( f(x) = -x - 11 \):
\( f(-9) = -(-9) - 11 = 9 - 11 = -2 \)

Step 3: Calculate \( -5f(12) - 2f(-9) \)

Substitute \( f(12) = -23 \) and \( f(-9) = -2 \):
\( -5(-23) - 2(-2) = 115 + 4 = 119 \)

4) \( \frac{f(9)}{f(-6)} \)

Step 1: Find \( f(9) \)

Substitute \( x = 9 \) into \( f(x) = -x - 11 \):
\( f(9) = -9 - 11 = -20 \)

Step 2: Find \( f(-6) \)

Substitute \( x = -6 \) into \( f(x) = -x - 11 \):
\( f(-6) = -(-6) - 11 = 6 - 11 = -5 \)

Step 3: Calculate \( \frac{f(9)}{f(-6)} \)

Substitute \( f(9) = -20 \) and \( f(-6) = -5 \):
\( \frac{-20}{-5} = 4 \)

Final Answers:

1. \(\boxed{-96}\)
2. \(\boxed{624}\)
3. \(\boxed{119}\)
4. \(\boxed{4}\)

Answer:

To solve these problems, we use the function \( f(x) = -x - 11 \) and substitute the given values of \( x \) into the function.

1) \( 4f(-2) + 5f(1) \)

Step 1: Find \( f(-2) \)

Substitute \( x = -2 \) into \( f(x) = -x - 11 \):
\( f(-2) = -(-2) - 11 = 2 - 11 = -9 \)

Step 2: Find \( f(1) \)

Substitute \( x = 1 \) into \( f(x) = -x - 11 \):
\( f(1) = -1 - 11 = -12 \)

Step 3: Calculate \( 4f(-2) + 5f(1) \)

Substitute \( f(-2) = -9 \) and \( f(1) = -12 \):
\( 4(-9) + 5(-12) = -36 - 60 = -96 \)

2) \( 3f(5) \times f(2) \)

Step 1: Find \( f(5) \)

Substitute \( x = 5 \) into \( f(x) = -x - 11 \):
\( f(5) = -5 - 11 = -16 \)

Step 2: Find \( f(2) \)

Substitute \( x = 2 \) into \( f(x) = -x - 11 \):
\( f(2) = -2 - 11 = -13 \)

Step 3: Calculate \( 3f(5) \times f(2) \)

Substitute \( f(5) = -16 \) and \( f(2) = -13 \):
\( 3(-16) \times (-13) = -48 \times (-13) = 624 \)

3) \( -5f(12) - 2f(-9) \)

Step 1: Find \( f(12) \)

Substitute \( x = 12 \) into \( f(x) = -x - 11 \):
\( f(12) = -12 - 11 = -23 \)

Step 2: Find \( f(-9) \)

Substitute \( x = -9 \) into \( f(x) = -x - 11 \):
\( f(-9) = -(-9) - 11 = 9 - 11 = -2 \)

Step 3: Calculate \( -5f(12) - 2f(-9) \)

Substitute \( f(12) = -23 \) and \( f(-9) = -2 \):
\( -5(-23) - 2(-2) = 115 + 4 = 119 \)

4) \( \frac{f(9)}{f(-6)} \)

Step 1: Find \( f(9) \)

Substitute \( x = 9 \) into \( f(x) = -x - 11 \):
\( f(9) = -9 - 11 = -20 \)

Step 2: Find \( f(-6) \)

Substitute \( x = -6 \) into \( f(x) = -x - 11 \):
\( f(-6) = -(-6) - 11 = 6 - 11 = -5 \)

Step 3: Calculate \( \frac{f(9)}{f(-6)} \)

Substitute \( f(9) = -20 \) and \( f(-6) = -5 \):
\( \frac{-20}{-5} = 4 \)

Final Answers:

1. \(\boxed{-96}\)
2. \(\boxed{624}\)
3. \(\boxed{119}\)
4. \(\boxed{4}\)