Question
Choose the statement that best describes a true relationship between any point on f (*) and
-f（一z）. （1 point）
O it (z, y) is on the graph of f|(a), then (I, - y) is on the graph of - / (-*).
O it (z, y) is on the graph of f (z), then (-7, - y) is on the graph of - / (-*).
• i (z, y) is on the graph of J (a), then (z, y) is on the graph of - / (-z)
O if (z, y) is on the graph of / (z), then (- 2, y) is on the graph of -f(-z).
-f（一z）. （1 point）
O it (z, y) is on the graph of f|(a), then (I, - y) is on the graph of - / (-*).
O it (z, y) is on the graph of f (z), then (-7, - y) is on the graph of - / (-*).
• i (z, y) is on the graph of J (a), then (z, y) is on the graph of - / (-z)
O if (z, y) is on the graph of / (z), then (- 2, y) is on the graph of -f(-z).
Answers
GPT-4o mini
To determine the correct statement about the relationship between the function \( f(z) \) and the function \( -f(-z) \), let’s examine what happens to the coordinates when we apply these transformations.
1. **Understanding the transformations**:
- If \( (z, y) \) is a point on the graph of \( f(z) \), it means \( y = f(z) \).
- The expression \( -f(-z) \) reflects the function in the y-direction and then in the x-direction (it flips both axes).
2. **Evaluating the statements**:
- **Statement 1**: If \( (z, y) \) is on the graph of \( f(z) \), then \( (z, -y) \) is on the graph of \( -f(-z) \).
- This is **false**. The point \( (z, y) \) translates to \( (-z, -y) \).
- **Statement 2**: If \( (z, y) \) is on the graph of \( f(z) \), then \( (-z, -y) \) is on the graph of \( -f(-z) \).
- This is **true**. From the above discussion, if \( y = f(z) \), then \( -y = -f(-z) \).
- **Statement 3**: If \( (z, y) \) is on the graph of \( f(z) \), then \( (z, y) \) is on the graph of \( -f(-z) \).
- This is **false**, as we aren’t getting the same point.
- **Statement 4**: If \( (z, y) \) is on the graph of \( f(z) \), then \( (-2, y) \) is on the graph of \( -f(-z) \).
- This is **false**; the transformation does not change the x-coordinate independently to -2.
Given the analysis, the **correct answer** is:
**If \( (z, y) \) is on the graph of \( f(z) \), then \( (-z, -y) \) is on the graph of \( -f(-z) \).**
1. **Understanding the transformations**:
- If \( (z, y) \) is a point on the graph of \( f(z) \), it means \( y = f(z) \).
- The expression \( -f(-z) \) reflects the function in the y-direction and then in the x-direction (it flips both axes).
2. **Evaluating the statements**:
- **Statement 1**: If \( (z, y) \) is on the graph of \( f(z) \), then \( (z, -y) \) is on the graph of \( -f(-z) \).
- This is **false**. The point \( (z, y) \) translates to \( (-z, -y) \).
- **Statement 2**: If \( (z, y) \) is on the graph of \( f(z) \), then \( (-z, -y) \) is on the graph of \( -f(-z) \).
- This is **true**. From the above discussion, if \( y = f(z) \), then \( -y = -f(-z) \).
- **Statement 3**: If \( (z, y) \) is on the graph of \( f(z) \), then \( (z, y) \) is on the graph of \( -f(-z) \).
- This is **false**, as we aren’t getting the same point.
- **Statement 4**: If \( (z, y) \) is on the graph of \( f(z) \), then \( (-2, y) \) is on the graph of \( -f(-z) \).
- This is **false**; the transformation does not change the x-coordinate independently to -2.
Given the analysis, the **correct answer** is:
**If \( (z, y) \) is on the graph of \( f(z) \), then \( (-z, -y) \) is on the graph of \( -f(-z) \).**