Question

homework assignment 1.1: functions and function notation
score: 7/15 answered: 7/15
question 8
suppose $f(x) = -2x^2 - 7x + 7$. compute the following:
a.) $f(-1) + f(5) = $
b.) $f(-1) - f(5) = $
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Explanation:

Part A: Compute \( f(-1) + f(5) \)

Step 1: Calculate \( f(-1) \)

Substitute \( x = -1 \) into \( f(x) = -2x^2 - 7x + 7 \):
\[

$$\begin{align*} f(-1) &= -2(-1)^2 - 7(-1) + 7 \\ &= -2(1) + 7 + 7 \\ &= -2 + 7 + 7 \\ &= 12 \end{align*}$$

\]

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

Substitute \( x = 5 \) into \( f(x) = -2x^2 - 7x + 7 \):
\[

$$\begin{align*} f(5) &= -2(5)^2 - 7(5) + 7 \\ &= -2(25) - 35 + 7 \\ &= -50 - 35 + 7 \\ &= -78 \end{align*}$$

\]

Step 3: Compute \( f(-1) + f(5) \)

Add the results from Step 1 and Step 2:
\[
f(-1) + f(5) = 12 + (-78) = -66
\]

Part B: Compute \( f(-1) - f(5) \)

Step 1: Use results from Part A

We already know \( f(-1) = 12 \) and \( f(5) = -78 \).

Step 2: Compute \( f(-1) - f(5) \)

Subtract the results:
\[
f(-1) - f(5) = 12 - (-78) = 12 + 78 = 90
\]

Final Answers

A. \(\boldsymbol{-66}\)
B. \(\boldsymbol{90}\)

Answer:

Part A: Compute \( f(-1) + f(5) \)

Step 1: Calculate \( f(-1) \)

Substitute \( x = -1 \) into \( f(x) = -2x^2 - 7x + 7 \):
\[

$$\begin{align*} f(-1) &= -2(-1)^2 - 7(-1) + 7 \\ &= -2(1) + 7 + 7 \\ &= -2 + 7 + 7 \\ &= 12 \end{align*}$$

\]

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

Substitute \( x = 5 \) into \( f(x) = -2x^2 - 7x + 7 \):
\[

$$\begin{align*} f(5) &= -2(5)^2 - 7(5) + 7 \\ &= -2(25) - 35 + 7 \\ &= -50 - 35 + 7 \\ &= -78 \end{align*}$$

\]

Step 3: Compute \( f(-1) + f(5) \)

Add the results from Step 1 and Step 2:
\[
f(-1) + f(5) = 12 + (-78) = -66
\]

Part B: Compute \( f(-1) - f(5) \)

Step 1: Use results from Part A

We already know \( f(-1) = 12 \) and \( f(5) = -78 \).

Step 2: Compute \( f(-1) - f(5) \)

Subtract the results:
\[
f(-1) - f(5) = 12 - (-78) = 12 + 78 = 90
\]

Final Answers

A. \(\boldsymbol{-66}\)
B. \(\boldsymbol{90}\)