25. Let f be a function such that f(x) = f(-x) for all real numbers x. If the point (-2, 4) lies on the graph of y = f(x) in the xy-plane, which of the following points must also lie on the graph of y = f(x)?
Answer: E
The point (2, 4) must also lie on the graph of y = f(x).
Since the function f satisfies the property f(x) = f(-x), it is an even function. Given that the point (-2, 4) lies on the graph, it follows that f(-2) = 4, and thus f(2) must also equal 4, meaning the point (2, 4) must be on the graph.
A) (-2, -4)
This option is incorrect because the function is even, which means that if (-2, 4) is on the graph, then its corresponding point (2, 4) must also be on the graph. There is no indication from the information provided that f(-2) would equal -4.
B) (0, 0)
This option is not necessarily correct as the information given does not imply that the function must pass through the origin. The value of f(0) is not determined by the information about the point (-2, 4).
C) (0, 4)
This option is also incorrect as it does not relate to the even function property established by the point (-2, 4). The function's behavior at x = 0 is not determined by the points provided.
D) (2, -4)
This choice is incorrect because, although it is a reflection across the y-axis of (-2, -4), the function is even and we have established that f(2) must equal 4, not -4.
E) (2, 4)
This option is correct because the even nature of the function f ensures that if f(-2) = 4, then f(2) must also equal 4. Therefore, the point (2, 4) must lie on the graph of y = f(x).
Conclusion
The point (2, 4) is definitively the correct answer as it arises directly from the property of even functions. All other options fail to satisfy the criteria set by the function's properties or the given point, reinforcing that (2, 4) is the only valid conclusion based on the information provided.