Ch. 1 - Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 1 - Functions

Problem 1.2.14a

Chapter 1, Problem 1.2.14a

Composition of Functions

Copy and complete the following table.

a. <IMAGE>

Verified step by step guidance

Step 1: Understand the concept of composition of functions. The composition of two functions f and g, denoted as (f ∘ g)(x), means applying g first and then applying f to the result of g(x).

Step 2: Identify the functions involved in the composition from the table or image provided. Let's assume f(x) and g(x) are given functions.

Step 3: For each x value in the table, calculate g(x) first. This involves substituting the x value into the function g.

Step 4: Use the result from Step 3 as the input for the function f. Substitute g(x) into f to find f(g(x)).

Step 5: Record the result of f(g(x)) for each x value in the table to complete the composition of functions table.

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Key Concepts

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

Composition of Functions

The composition of functions involves combining two functions to create a new function. If you have two functions, f(x) and g(x), the composition is denoted as (f ∘ g)(x) = f(g(x)). This means you first apply g to x, and then apply f to the result of g. Understanding this concept is crucial for manipulating and evaluating complex functions.

Function Notation

Function notation is a way to represent functions mathematically, typically using symbols like f(x) to denote the output of a function f for a given input x. This notation allows for clear communication of mathematical ideas and operations involving functions, making it easier to work with compositions and transformations of functions.

Evaluating Functions

Evaluating functions involves substituting a specific value into a function to find the corresponding output. For example, if f(x) = 2x + 3, evaluating f(2) would yield 2(2) + 3 = 7. This skill is essential for working with composed functions, as it requires substituting values into multiple functions in sequence.

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Evaluating Composed Functions

Related Practice

Textbook Question

Piecewise-Defined Functions

Find a formula for each function graphed in Exercises 29–32.

a. <IMAGE>

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Textbook Question

Composition of Functions

Let f(x) = x − 3, g(x) = √x, h(x) = x³, and j(x) = 2x. Express each of the functions in Exercises 11 and 12 as a composition involving one or more of f, g, h, and j.

a. y = 2x − 3

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Textbook Question

Industrial costs A power plant sits next to a river where the river is 800 ft wide. Laying a new cable from the plant to a location in the city 2 mi downstream on the opposite side costs \$180 per foot across the river and \$100 per foot along the land.

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b. Generate a table of values to determine whether the least expensive location for point Q is less than 2000 ft or greater than 2000 ft from point P.

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Textbook Question

Functions and Graphs

Graph the following equations and explain why they are not graphs of functions of x.

a. |x| + |y| = 1

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Textbook Question

Piecewise-Defined Functions

Find a formula for each function graphed in Exercises 29–32.

b. <IMAGE>