Final answer:
Option 1, f(x) = x^2 and Option 3, f(x) = cos(x), are the even functions from the given options. This conclusion is based on the definition of even functions, which are symmetric about the y-axis.
Step-by-step explanation:
In mathematics, particularly in the study of functions, we often distinguish between even and odd functions based on their symmetry properties. An even function is symmetric about the y-axis, meaning f(x) = f(-x) for all x in the domain of f. An odd function, on the other hand, has rotational symmetry with respect to the origin, which can be formally stated as f(-x) = -f(x) for all x in the domain of f.
Given the options provided, we can use these definitions to determine which functions are even:
• Option 1: f(x) = x^2. This is an even function because (-x)^2 = x^2.
• Option 2: f(x) = sin(x). This is not an even function because sin(-x) ≠ sin(x). However, it is worth noting that sine is an odd function because sin(-x) = -sin(x).
• Option 3: f(x) = cos(x). This is an even function, as cos(-x) = cos(x).
• Option 4: f(x) = x^3. This function is not even because (-x)^3 = -x^3 which is not equal to x^3. It is an odd function.
Upon reviewing the functions, we can conclude that Options 1 and 3, which are f(x) = x^2 and f(x) = cos(x), are even functions.