Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 1 - Functions

Problem 88c

Chapter 1, Problem 88c

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))

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Textbook Question

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>

g. ƒ (g(g(-2)))

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Textbook Question

Composition of even and odd functions from graphs Assume ƒ is an even function and g is an odd function. Use the (incomplete) graphs of ƒ and g in the figure to determine the following function values. <IMAGE>

a. ƒ(g(-2))

i. g(g(g(-1)))

a. ƒ(g(-1))

e. g(g(-1))

e. g(g(-7))