To show that \( h(x) \) is the reflection about the origin of the function \( f(x) \), we need to find \( h(x) \) such that \( h(x) = -f(-x) \).
Let's fill in the table for \( h(x) \) based on the values of \( f(x) \):
| x | f(x) | h(x) | |------|--------|--------| | 0 | 0 | 0 | | 1 | 1 | -1 | | 2 | 8 | -8 |
Explanation:
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For \( x = 0 \):
- \( f(0) = 0 \)
- \( h(0) = -f(-0) = -f(0) = -0 = 0 \)
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For \( x = 1 \):
- \( f(1) = 1 \)
- \( h(1) = -f(-1) \) (need to find \( f(-1) \); if it’s not provided, we cannot calculate).
However, based on the reflection about the origin concept, it's typically \(-f(1)\):
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\( h(1) = -f(1) = -1 \)
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For \( x = 2 \):
- \( f(2) = 8 \)
- \( h(2) = -f(-2) \) (again, need \( f(-2) \) value).
Just for reflection:
- \( h(2) = -f(2) = -8 \)
We provided values for \( h(x) \) based on the pattern of reflection across the origin, which demonstrates that \( h(x) \) reflects \( f(x) \) appropriately. If you have specific values for \( f(-1) \) and \( f(-2) \), please provide, and we can adjust accordingly.