To understand the relationship between \( f(x) \) and \( f(-x) \), let's analyze the implications:
- If \( (x, y) \) is on the graph of \( f(x) \), that means \( y = f(x) \).
- For the point \( (-x, y) \), to find the corresponding y-value when plugging in \( -x \) into \( f \), we have: \( f(-x) \).
So we know:
- The point \( (x, y) \) corresponds to \( (x, f(x)) \).
- The point \( (-x, y) \) would correspond to \( (-x, f(-x)) \).
Now let's analyze the statements one by one:
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If (x, y) is on the graph of f(x), then (x, -y) is on the graph of f(-x).
This is incorrect because it misrepresents the relationship. If \( y = f(x) \), then \( -y \) is not related to \( f(-x) \). -
If (x, y) is on the graph of f(x), then (-x, -y) is on the graph of f(-x).
This is also incorrect. If \( y = f(x) \), then we would have \( -y = -f(x) \) and not necessarily equal to \( f(-x) \). -
If (x, y) is on the graph of f(x), then (x, y) is on the graph of f(-x).
This is incorrect since substituting \( x \) does not generate the same values in \( f(-x) \). -
If (x, y) is on the graph of f(x), then (-x, y) is on the graph of f(-x).
This is incorrect. Instead, for a point \( (x, f(x)) \), the corresponding point at \( -x \) would yield \( (-x, f(-x)) \).
None of the provided statements accurately represents the true mathematical relationship between \( f(x) \) and \( f(-x) \).
The correct relationship would be:
- If \( (x, y) \) is on the graph of \( f(x) \), then \( (-x, f(-x)) \) correlates correctly, reflecting the function's output at \( -x \).
So, technically, none of the statements provided are correct with respect to describing the relationship between the functions properly. The correct reflection would be to state that if \( (x, y) \) is on the graph of \( f(x) \), then \( (-x, f(-x)) \) is the correct point on the graph of \( f(-x) \).
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