Question

Finish creating the input-output table for the function f(x)=−(x+1)2+4 . Which of the given x values produces the largest output for the function?

Answer 1

To create the input-output table for the function \( f(x) = -(x + 1)^2 + 4 \), we need to calculate the output for various values of \( x \). We can choose a set of \( x \) values to evaluate. Let's use \( x = -3, -2, -1, 0, 1, 2 \) as our input values.

Now, we will calculate \( f(x) \) for each of these \( x \) values:

1. For \( x = -3 \):
\[
f(-3) = -((-3) + 1)^2 + 4 = -(-2)^2 + 4 = -4 + 4 = 0
\]

2. For \( x = -2 \):
\[
f(-2) = -((-2) + 1)^2 + 4 = -(-1)^2 + 4 = -1 + 4 = 3
\]

3. For \( x = -1 \):
\[
f(-1) = -((-1) + 1)^2 + 4 = -(0)^2 + 4 = 0 + 4 = 4
\]

4. For \( x = 0 \):
\[
f(0) = -(0 + 1)^2 + 4 = -(1)^2 + 4 = -1 + 4 = 3
\]

5. For \( x = 1 \):
\[
f(1) = -(1 + 1)^2 + 4 = -(2)^2 + 4 = -4 + 4 = 0
\]

6. For \( x = 2 \):
\[
f(2) = -(2 + 1)^2 + 4 = -(3)^2 + 4 = -9 + 4 = -5
\]

Now, we can create the input-output table:

| \( x \) | \( f(x) \) |
|---------|------------|
| -3 | 0 |
| -2 | 3 |
| -1 | 4 |
| 0 | 3 |
| 1 | 0 |
| 2 | -5 |

From the table, we can see that \( f(-1) = 4 \) is the largest output value. Therefore, the \( x \) value that produces the largest output for the function is:

- \( x = -1 \) produces the largest output of \( 4 \).