Composition of even and odd functions from tables Assume ƒ is an even function, g is an odd function, and both are defined at 0. Use the (incomplete) table to evaluate the given compositions. <IMAGE>

c. ƒ(g(-3))

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Identify the properties of even and odd functions: An even function satisfies \( f(x) = f(-x) \) and an odd function satisfies \( g(x) = -g(-x) \).

Since \( g \) is an odd function, calculate \( g(-3) \) using the property \( g(-3) = -g(3) \).

Look up the value of \( g(3) \) in the table provided. If \( g(3) \) is not directly available, use any given information to deduce it.

Substitute the value of \( g(-3) \) into the function \( f \) to find \( f(g(-3)) \).

Since \( f \) is an even function, use the property \( f(x) = f(-x) \) if needed to simplify the evaluation of \( f(g(-3)) \).

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주요 개념

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Even Functions

An even function is defined by the property that f(x) = f(-x) for all x in its domain. This symmetry about the y-axis means that the function's values are the same for both positive and negative inputs. For example, the function f(x) = x² is even because f(2) = f(-2) = 4.

Odd Functions

An odd function satisfies the condition g(x) = -g(-x) for all x in its domain. This property indicates that the function is symmetric about the origin, meaning that if you reflect the graph across both axes, it remains unchanged. A classic example is g(x) = x³, where g(2) = 8 and g(-2) = -8.

Function Composition

Function composition involves combining two functions where the output of one function becomes the input of another. Denoted as (f ∘ g)(x) = f(g(x)), this operation requires evaluating the inner function first, followed by the outer function. In the context of the question, evaluating f(g(-3)) means first finding g(-3) and then using that result as the input for f.

Composition of even and odd functions from tables Assume ƒ is an even function, g is an odd function, and both are defined at 0. Use the (incomplete) table to evaluate the given compositions. <IMAGE>c. ƒ(g(-3))