Question
Complete the table to show that h of X is the reflection about the origin of the function f of x x 0 1 2 f of x 0 1 8 x 0 8 of x 0
Answers
GPT-4o mini
To show that \( h(x) \) is the reflection about the origin of the function \( f(x) \), we need to understand what reflection about the origin means. The reflection about the origin transforms a point \( (x, y) \) to \( (-x, -y) \). Thus, the relationship between \( f(x) \) and \( h(x) \) can be expressed as:
\[
h(x) = -f(-x)
\]
From the information provided, let's clarify the values of the function \( f(x) \). It seems like you have provided some points for \( f(x) \):
- \( f(0) = 1 \)
- \( f(1) = 8 \)
- \( f(2) = 0 \)
Now we need to calculate \( h(x) \) using the transformation \( h(x) = -f(-x) \).
### Step-by-Step Calculation
1. **Find \( f(-x) \)**:
- \( f(0) \) corresponds to \( x = 0 \).
- For \( x = 1 \), \( -x = -1 \) so we need to calculate \( f(-1) \) (which is not provided in your data).
- For \( x = 2 \), \( -x = -2 \), so we also need \( f(-2) \) (not provided).
- Let's assume \( f(-1) \) and \( f(-2) \) have some values or we would just leave these as unknowns.
2. **Evaluating \( h(x) \)**:
- \( h(0) = -f(0) = -1 \)
- Since we do not have values for \( f(-1) \) and \( f(-2) \), we leave them as unknowns:
- \( h(1) = -f(-1) \) (unknown)
- \( h(2) = -f(-2) \) (unknown)
### Complete Table Based on Current Data
| \( x \) | \( f(x) \) | \( h(x) \) |
|---------|------------|-------------|
| 0 | 1 | -1 |
| 1 | 8 | -f(-1) |
| 2 | 0 | -f(-2) |
To fully complete the table, more data for \( f(-1) \) and \( f(-2) \) would be needed, or you may need to provide specific values if they were part of the context.
If you have additional details regarding the function \( f(x) \) at those negative values or the context in which this function operates, please share that for a more complete analysis.
\[
h(x) = -f(-x)
\]
From the information provided, let's clarify the values of the function \( f(x) \). It seems like you have provided some points for \( f(x) \):
- \( f(0) = 1 \)
- \( f(1) = 8 \)
- \( f(2) = 0 \)
Now we need to calculate \( h(x) \) using the transformation \( h(x) = -f(-x) \).
### Step-by-Step Calculation
1. **Find \( f(-x) \)**:
- \( f(0) \) corresponds to \( x = 0 \).
- For \( x = 1 \), \( -x = -1 \) so we need to calculate \( f(-1) \) (which is not provided in your data).
- For \( x = 2 \), \( -x = -2 \), so we also need \( f(-2) \) (not provided).
- Let's assume \( f(-1) \) and \( f(-2) \) have some values or we would just leave these as unknowns.
2. **Evaluating \( h(x) \)**:
- \( h(0) = -f(0) = -1 \)
- Since we do not have values for \( f(-1) \) and \( f(-2) \), we leave them as unknowns:
- \( h(1) = -f(-1) \) (unknown)
- \( h(2) = -f(-2) \) (unknown)
### Complete Table Based on Current Data
| \( x \) | \( f(x) \) | \( h(x) \) |
|---------|------------|-------------|
| 0 | 1 | -1 |
| 1 | 8 | -f(-1) |
| 2 | 0 | -f(-2) |
To fully complete the table, more data for \( f(-1) \) and \( f(-2) \) would be needed, or you may need to provide specific values if they were part of the context.
If you have additional details regarding the function \( f(x) \) at those negative values or the context in which this function operates, please share that for a more complete analysis.