To describe the relationship between any point on \( f(x) \) and \( f(-x) \), we can focus on the definition of the function and its transformation.
Letβs analyze the options provided:
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If \( (x, y) \) is on the graph of \( f(x) \), then \( (x, -y) \) is on the graph of \( f(-x) \).
- Incorrect: This describes a reflection across the x-axis, which is not what the transformation from \( f(x) \) to \( f(-x) \) does.
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If \( (x, y) \) is on the graph of \( f(x) \), then \( (x, -y) \) is on the graph of \( f(-x) \).
- Incorrect: This also suggests a reflection that is not relevant to the transformation we are discussing.
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If \( (x, y) \) is on the graph of \( f(x) \), then \( (-x, y) \) is on the graph of \( f(-x) \).
- Correct: This correctly states that reflecting a point over the y-axis results in a point on the graph of \( f(-x) \).
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If \( (x, y) \) is on the graph of \( f(x) \), then \( (-x, y) \) is on the graph of \( f(-x) \).
- Correct: This statement is actually a repetition of the previous option (misspelled twice) but it conveys the same true reflection.
Thus, the correct statement to describe the true relationship is:
If \( (x, y) \) is on the graph of \( f(x) \), then \( (-x, y) \) is on the graph of \( f(-x) \).