Rationale
(2, 4)
The function ( f(x) = f(-x) ) indicates that ( f ) is an even function, meaning the outputs for ( x ) and ( -x ) are identical. Given that the point (-2, 4) lies on the graph, it follows that the point (2, 4) must also be on the graph, confirming the even nature of ( f ).
A) (-2, -4)
This point does not satisfy the property of the function. Since ( f(-2) = 4 ), it cannot be the case that ( f(-2) = -4 ). The point (-2, -4) would imply that the function takes on a different value for ( -2 ), contradicting the definition of an even function.
B) (0, 0)
While this point is relevant for many functions, it does not necessarily follow from the given information. The equation ( f(x) = f(-x) ) does not imply that ( f(0) = 0 ). The function could take any value at 0, and we have no information to conclude that (0, 0) must be on the graph.
C) (0, 4)
Similar to choice B, the function might or might not equal 4 when ( x = 0 ). The evenness of the function does not dictate the specific value at ( x = 0). Therefore, we cannot conclude that (0, 4) is a point on the graph.
D) (2, -4)
This point also contradicts the property of the function. If ( f(2) = -4 ), then ( f(-2) ) would need to equal -4, which is inconsistent with the known point (-2, 4). As an even function, ( f(-2) ) must equal 4, ruling out (2, -4).
Conclusion
Given that ( f(x) = f(-x) ), the point (2, 4) must lie on the graph of ( y = f(x) ) alongside the point (-2, 4). The even nature of the function guarantees that for every point (a, b), there's a corresponding point (-a, b), but not necessarily any other combinations of values. Thus, (2, 4) is the only point that must also be on the graph alongside (-2, 4).
