Ohio Assessments for Educators (OAE) Mathematics Practice Exam

Which of the following is characteristic of an even function?

1. The function is symmetric with respect to the origin

2. f(x) = -f(-x)

3. The function satisfies f(x) = f(-x)

4. The function has an odd degree

The correct answer is: The function satisfies f(x) = f(-x)

An even function is defined by the property that its output remains the same when the input is negated. This is mathematically expressed as f(x) = f(-x). This symmetry means that if you were to graph the function, it would be symmetric about the y-axis. For every point (x, f(x)) on the graph, there exists a corresponding point (-x, f(x)), indicating that the function behaves identically on both sides of the y-axis. In contrast, the other choices present different characteristics. The first choice describes an odd function, which is symmetric with respect to the origin, indicating that f(x) is equal to -f(-x). The second choice, which states f(x) = -f(-x), confirms this odd function property and does not apply to even functions. Lastly, the property of a function having an odd degree pertains to polynomial functions, but doesn't inherently define whether they're even or odd. Hence, the defining trait of an even function is its symmetry with respect to the y-axis, which aligns with the property fulfilled by f(x) = f(-x).