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>


a. ƒ(g(-1))

Verified step by step guidance

1. Identify the properties of even and odd functions: An even function satisfies \( f(x) = f(-x) \) and an odd function satisfies \( g(x) = -g(-x) \).
2. Since \( g \) is an odd function, calculate \( g(-1) \) using the property \( g(-1) = -g(1) \).
3. Look up the value of \( g(1) \) in the table to find \( g(-1) \).
4. Substitute the value of \( g(-1) \) into \( f(g(-1)) \).
5. Use the property of the even function \( f \), which is \( f(x) = f(-x) \), to evaluate \( f(g(-1)) \) using the table.

Key Concepts

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) = 4 and 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 understanding the individual functions' behaviors. In the context of the question, evaluating f(g(-1)) means first finding g(-1) and then applying f to that result.