What term describes a function that is symmetric with respect to the y-axis?

What term describes a function that is symmetric with respect to the y-axis?

• A. Odd function
• B. Even function
• C. Polynomial function
• D. Transcendental function

Answer: (B)

A function that is symmetric with respect to the y-axis is specifically defined as an even function. This means that for every point (x, f(x)) on the graph of the function, the point (-x, f(x)) is also included in the graph. The defining property of even functions is expressed mathematically as f(x) = f(-x) for all x in the function's domain. This symmetry indicates that the graph looks the same on both sides of the y-axis. In contrast, odd functions exhibit symmetry about the origin and fulfill the condition f(-x) = -f(x), which is distinct from the requirement for even functions. Polynomial functions can be either even, odd, or neither, depending on their specific degree and coefficients but do not inherently possess y-axis symmetry unless they meet the criteria for even functions. Transcendental functions do not fall into these categories clearly either; they are a broader class that includes functions such as exponentials and logarithms, which do not necessarily exhibit any symmetry. Thus, the correct terminology for describing a function with this particular symmetry is "even function."

A function that is symmetric with respect to the y-axis is specifically defined as an even function. This means that for every point (x, f(x)) on the graph of the function, the point (-x, f(x)) is also included in the graph. The defining property of even functions is expressed mathematically as f(x) = f(-x) for all x in the function's domain. This symmetry indicates that the graph looks the same on both sides of the y-axis.

In contrast, odd functions exhibit symmetry about the origin and fulfill the condition f(-x) = -f(x), which is distinct from the requirement for even functions. Polynomial functions can be either even, odd, or neither, depending on their specific degree and coefficients but do not inherently possess y-axis symmetry unless they meet the criteria for even functions. Transcendental functions do not fall into these categories clearly either; they are a broader class that includes functions such as exponentials and logarithms, which do not necessarily exhibit any symmetry. Thus, the correct terminology for describing a function with this particular symmetry is "even function."