To choose the correct statement describing the relationship between any point on \( f(x) \) and \( f(-x) \), we need to consider what \( f(-x) \) means.
If \((x, y)\) is a point on the graph of \( f(x) \), then \( y = f(x) \).
Now, for \( f(-x) \):
- If we substitute \(-x\) into the function, we get the point \((-x, f(-x))\).
- Therefore, if we have a point \((x, y)\), when we consider this point in \( f(-x) \), we actually get \((-x, y)\) because \( y = f(x) \) does not change.
Looking at the responses:
-
If (x, y) is on the graph of f(x), then (x, -y) is on the graph of f(-x): This is incorrect because \((x, y)\) on \( f(x) \) does not imply that \((x, -y)\) relates to \( f(-x) \).
-
If (x, y) is on the graph of f(x), then (-x, -y) is on the graph of f(-x): This statement is also incorrect because it misrepresents the transformation. The point \((-x, -y)\) is not directly related to the point \((x, y)\) regarding \( f(-x) \).
Thus, neither statement is correct in depicting the relationship between points on the graphs of \( f(x) \) and \( f(-x) \).
However, the correct relationship is if \( (x, y) \) is on the graph of \( f(x) \), then \( (-x, f(-x)) \) is on the graph of \( f(-x) \). If you meant to characterize \( f(x) \) and \( f(-x) \) as merely opposite in terms of the x-value:
- No option is accurately capturing the relationship described above.
You may want to consider re-evaluating the provided statements or producing new options that more closely align with the mathematical properties of functions and symmetry relative to the y-axis.
1 answer