As an expert in the field of mathematics, particularly in the study of functions and their properties, I can provide a comprehensive explanation regarding the symmetry of functions and the axis about which an even function is reflected.

Even and Odd Functions:
In mathematics, even and odd functions are two types of functions that exhibit symmetry. Symmetry in this context refers to the way a function's graph appears in relation to the coordinate axes.

Even Functions:
An even function is defined as a function \( f(x) \) that satisfies the condition \( f(x) = f(-x) \) for all \( x \) in the domain of \( f \). This means that for every point \( (x, f(x)) \) on the graph of the function, there is a corresponding point \( (-x, f(x)) \) that also lies on the graph. This property is known as reflection symmetry about the y-axis, which is also referred to as the vertical axis of symmetry.

The reflection symmetry about the y-axis implies that the graph of an even function is unchanged when it is reflected across the y-axis. This is a visual way to understand the symmetry property of even functions. For instance, the functions \( f(x) = x^2 \) and \( f(x) = \cos(x) \) are both even because they satisfy the condition \( f(x) = f(-x) \) for all \( x \) in their respective domains.

Odd Functions:
Conversely, an odd function is a function \( f(x) \) that satisfies the condition \( f(-x) = -f(x) \) for all \( x \) in the domain of \( f \). This means that for every point \( (x, f(x)) \) on the graph of the function, there is a corresponding point \( (-x, -f(x)) \) that also lies on the graph. This property is known as rotational symmetry about the origin, which is also referred to as point symmetry or central symmetry.

The rotational symmetry about the origin implies that the graph of an odd function is rotated 180 degrees about the origin and coincides with the original graph. An example of an odd function is \( f(x) = x^3 \) or \( f(x) = \sin(x) \), as they satisfy the condition \( f(-x) = -f(x) \) for all \( x \) in their domains.

Determining Even or Odd Algebraically:
To determine if a function is even, odd, or neither algebraically, one can substitute \( -x \) for \( x \) in the function's formula and compute \( f(-x) \). If the result is \( f(-x) = f(x) \), then the function is even. If the result is \( f(-x) = -f(x) \), then the function is odd. If neither condition is met, the function is neither even nor odd.

Conclusion:
In summary, an even function is reflected over the y-axis, which is the vertical axis of symmetry. This reflection symmetry is a fundamental characteristic that defines even functions. Understanding the symmetry properties of functions is crucial for analyzing their behavior and solving various mathematical problems.

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An even function has reflection symmetry about the y-axis. An odd function has rotational symmetry about the origin. We can decide algebraically if a function is even, odd or neither by replacing x by -x and computing f(-x). If f(-x) = f(x), the function is even.read more >>