Assume that p is a function that is defined for all x. (a)...

Assume that $p$ is a function that is defined for all $x$.
(a) Prove that if $f$ is defined by $f(x)=p(x)+p(-x)$ then $f$ is even.
(b) Prove that if $g$ is defined by $g(x)=p(x)-p(-x)$ then $g$ is odd.

Key Concepts

Even Functions

An even function is defined as a function f that satisfies the condition f(-x) = f(x) for every x in its domain. This property implies that the graph of an even function is symmetric with respect to the y-axis. In proofs involving even functions, one typically demonstrates this symmetry by replacing x with -x and showing that the resulting expression is identical to the original function.

Odd Functions

An odd function is one that meets the condition f(-x) = -f(x) for every x in its domain. This definition means that the graph of an odd function exhibits rotational symmetry about the origin; that is, if you rotate the graph 180 degrees around the origin, it will look the same. Proofs of odd function properties involve substituting -x into the function and simplifying to show that it is the negative of the original function.

Function Transformation and Substitution

A key technique in analyzing functions, particularly when determining symmetry, is the substitution of -x for x in the function's definition. This transformation allows one to directly apply the definitions of even and odd functions. By evaluating the function at -x and comparing it to either f(x) or -f(x), one can verify the function's symmetry properties.

Algebraic Manipulation in Function Analysis

Algebraic operations, such as addition and subtraction, play a crucial role in the analysis of functions' symmetry properties. When combining the values of a function at x and -x, these operations can either highlight or cancel out certain elements, yielding an expression that clearly reflects even or odd symmetry. This approach is fundamental in proofs that use the definitions of even and odd functions.

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