To analyze the relationship between points on the graph of \( f(x) \) and the function \( -f(-x) \), let's start with any point \( (x, y) \) on the graph of \( f(x) \), where \( y = f(x) \).
Now, we want to determine the corresponding point on the graph of \( -f(-x) \):
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First, we evaluate \( -f(-x) \):
- If \( x \) corresponds to the input of \( f(x) \), then the input for \( -f(-x) \) will be \( -x \).
- The output will thus be \( -f(-x) \).
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The point \( (x, y) \) means \( y = f(x) \).
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To find the corresponding point for \( -f(-x) \), we can substitute \( -x \) into \( f \):
- This gives us \( y' = f(-x) \), resulting in the point \( (-x, f(-x)) \).
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We then take the negative of this output for our transformation:
- This gives \( -f(-x) \) resulting in the point \( (-x, -f(-x)) \).
So we have established that if \( (x, y) \) is on the graph of \( f(x) \), then the corresponding point on the graph of \( -f(-x) \) will be \( (-x, -y) \).
Thus, the statement that best describes the relationship is:
If (x,y) is on the graph of f(x), then (−x,−y) is on the graph of −f(−x).