Question
Complete the table to show that h(x)
is the reflection about the origin of the function f(x)
x f(x)
0 0
1 1
2 8
is the reflection about the origin of the function f(x)
x f(x)
0 0
1 1
2 8
Answers
GPT-4o mini
To show that \( h(x) \) is the reflection about the origin of the function \( f(x) \), we need to understand the property of reflections across the origin. For a function \( f(x) \), its reflection about the origin is represented by \( h(x) = -f(-x) \).
Let's compute the values for \( h(x) \) using the values of \( f(x) \) from your table and reflecting them about the origin.
We have the following points for \( f(x) \):
- \( f(0) = 0 \)
- \( f(1) = 1 \)
- \( f(2) = 8 \)
Now, we'll calculate \( h(x) \) for the corresponding negative values of \( x \):
### Step 1: Calculate \( f(-x) \)
- For \( x = 0 \):
- \( f(-0) = f(0) = 0 \)
- For \( x = 1 \):
- \( f(-1) \) is not provided directly, but we can use \( x = 2 \) next. We need this point later.
- For \( x = 2 \):
- \( f(-2) \) is not provided directly. We'll focus on \( x = -1 \) and \( x = -2 \) next.
### Step 2: Reflect and Fill the Table
Now we can fill in \( h(x) \) values for the \( x \) values of interest:
- For \( x = 0 \):
- \( h(0) = -f(0) = -0 = 0 \)
- For \( x = -1 \) (we need \( f(-1) \)):
Take an assumption or arbitrary value if needed. If we say \( f(-1) = 1 \) then:
- \( h(-1) = -f(1) = -1\)
- For \( x = -2 \):
Also not provided; let's say \( f(-2) = 8 \):
- \( h(-2) = -f(2) = -8\)
### Final Table
So we have:
| x | f(x) | h(x) |
|-----|-------|-------|
| 0 | 0 | 0 |
| 1 | 1 | -1 |
| 2 | 8 | -8 |
| -1 | | -1 |
| -2 | | -8 |
### Conclusion
With this setup, it demonstrates that \( h(x) \) is indeed the reflection about the origin of \( f(x) \) given that each value of \( h(x) \) corresponds to the negative of a value of \( f(x) \) evaluated at the negative input.
Make sure to verify the assumptions that were made about the values of \( f(-1) \) and \( f(-2) \) according to the context of your specific problem. If you have further information or more values for \( f(x) \), update accordingly!
Let's compute the values for \( h(x) \) using the values of \( f(x) \) from your table and reflecting them about the origin.
We have the following points for \( f(x) \):
- \( f(0) = 0 \)
- \( f(1) = 1 \)
- \( f(2) = 8 \)
Now, we'll calculate \( h(x) \) for the corresponding negative values of \( x \):
### Step 1: Calculate \( f(-x) \)
- For \( x = 0 \):
- \( f(-0) = f(0) = 0 \)
- For \( x = 1 \):
- \( f(-1) \) is not provided directly, but we can use \( x = 2 \) next. We need this point later.
- For \( x = 2 \):
- \( f(-2) \) is not provided directly. We'll focus on \( x = -1 \) and \( x = -2 \) next.
### Step 2: Reflect and Fill the Table
Now we can fill in \( h(x) \) values for the \( x \) values of interest:
- For \( x = 0 \):
- \( h(0) = -f(0) = -0 = 0 \)
- For \( x = -1 \) (we need \( f(-1) \)):
Take an assumption or arbitrary value if needed. If we say \( f(-1) = 1 \) then:
- \( h(-1) = -f(1) = -1\)
- For \( x = -2 \):
Also not provided; let's say \( f(-2) = 8 \):
- \( h(-2) = -f(2) = -8\)
### Final Table
So we have:
| x | f(x) | h(x) |
|-----|-------|-------|
| 0 | 0 | 0 |
| 1 | 1 | -1 |
| 2 | 8 | -8 |
| -1 | | -1 |
| -2 | | -8 |
### Conclusion
With this setup, it demonstrates that \( h(x) \) is indeed the reflection about the origin of \( f(x) \) given that each value of \( h(x) \) corresponds to the negative of a value of \( f(x) \) evaluated at the negative input.
Make sure to verify the assumptions that were made about the values of \( f(-1) \) and \( f(-2) \) according to the context of your specific problem. If you have further information or more values for \( f(x) \), update accordingly!