Create a table for the reflection about the origin of the function

Question

f(x) = (x + 2)} + 1.
Hint: First create a table of values for f (x) evaluated at x = 0, x = -1.,
and x = -2.

x - 0, 1, 2
-f(-x) - ?, ?, ?

Answer 1

To create a table for the reflection about the origin of the function \( f(x) = (x + 2) + 1 \), we first need to evaluate \( f(x) \) at the specified values of \( x = 0, -1, -2 \) and then find \( -f(-x) \) for those corresponding \( x \) values.

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

1. \( f(0) = (0 + 2) + 1 = 2 + 1 = 3 \)
2. \( f(-1) = (-1 + 2) + 1 = 1 + 1 = 2 \)
3. \( f(-2) = (-2 + 2) + 1 = 0 + 1 = 1 \)

Step 2: Calculate \( -f(-x) \)

Now, we will evaluate \( -f(-x) \):

1. For \( x = 0 \):

   • \( -f(-0) = -f(0) = -3 \)

2. For \( x = -1 \):

   • \( -f(-(-1)) = -f(1) \)
   • \( f(1) = (1 + 2) + 1 = 3 + 1 = 4 \)
   • So, \( -f(1) = -4 \)

3. For \( x = -2 \):

   • \( -f(-(-2)) = -f(2) \)
   • \( f(2) = (2 + 2) + 1 = 4 + 1 = 5 \)
   • So, \( -f(2) = -5 \)

Step 3: Create the Table

Now we can summarize the values in a table.

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

This table shows the function values and their corresponding reflections about the origin.