If f and g are both even functions, is f+g even? If f and g are both odd functions, is f+g odd?

step 1 This question gives us a series of three statements and asks us to determine if F plus G is even

If f and g are both even functions, is f+g even? If f and g...

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Key Concepts

Combination of Even and Odd Functions

When combining an even function with an odd function, the resulting function typically does not inherit the symmetry properties of either function. Evaluating the sum at -x does not yield a value that is either equal to the sum at x (which would make it even) or the negative of the sum at x (which would make it odd), unless one of the functions is identically zero. Therefore, the sum of an even and an odd function is generally neither even nor odd.

Odd Functions

An odd function is defined as one that satisfies the property f(-x) = -f(x) for all x in its domain. This reflects a symmetry about the origin, meaning that the function’s graph is invariant under a 180-degree rotation about the origin. The sum of two odd functions retains this symmetry, because applying the odd function property to the sum will show that it meets the condition for odd functions.

Even Functions

An even function is defined as one that satisfies the condition f(-x) = f(x) for all x in its domain. This property implies that the graph of the function is symmetric with respect to the y-axis. When adding two even functions, their symmetry about the y-axis is preserved, which is why the sum of two even functions is again even.

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