Assume that f is an even function, g is an odd function, and...

Assume that $f$ is an even function, $g$ is an odd function, and both $f$ and $g$ are defined on the entire real line $\mathbb{R} .$ Which of the following (where defined) are even? odd? a. $f g \quad$ b. $f / g \quad$ c. $g / f$ d. $f^{2}=f f \quad$ e. $g^{2}=g g \quad$ f. $f \circ g$ g. $g \circ f \quad$ h. $f \circ f \quad$ i. $g \circ g$

Assume that $f$ is an even function, $g$ is an odd function, and both $f$ and $g$ are defined on the entire real line $\mathbb{R} .$ Which of the following (where defined) are even? odd?
a. $f g \quad$ b. $f / g \quad$ c. $g / f$
d. $f^{2}=f f \quad$ e. $g^{2}=g g \quad$ f. $f \circ g$
g. $g \circ f \quad$ h. $f \circ f \quad$ i. $g \circ g$

Key Concepts

Product and Quotient of Functions

The product or quotient of functions involves combining functions through multiplication or division, respectively. An important aspect is understanding how the parity (evenness or oddness) of the combined functions affects the parity of the resulting function. For example, the product of an even and an odd function is generally odd, while the behavior of quotients depends on the relative properties and the domains of the functions involved.

Composition of Functions

Function composition involves applying one function to the result of another, denoted by (f ? g)(x) = f(g(x)). In the context of even and odd functions, understanding how the inner function's transformation of inputs interacts with the outer function's symmetry properties is essential. This concept helps determine whether the resulting composite function is even, odd, or neither.

Even Function

An even function is defined by the property that its value at negative inputs is identical to its value at positive inputs, i.e., f(-x) = f(x) for all real numbers. This symmetry about the y-axis is a fundamental concept when analyzing the behavior of functions under various operations such as addition, multiplication, and composition.

Odd Function

An odd function has the defining characteristic that its value at negative inputs is the negative of its value at positive inputs, meaning g(-x) = -g(x) for all real numbers. This antisymmetry about the origin plays a crucial role when evaluating products, quotients, and compositions of functions.

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