Ch. 1 - Functions

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

Hass 15th Edition

Ch. 1 - Functions

Problem 1.2.16c

Chapter 1, Problem 1.2.16c

Composition of Functions

Evaluate each expression using the functions
f(x) = 2 − x, g(x) = { −x, −2 ≤ x < 0
x − 1, 0 ≤ x ≤ 2

c. g(g(−1))

Verified step by step guidance

First, identify the function g(x) that applies to the input value x = -1. Since -1 falls within the interval -2 ≤ x < 0, use the expression g(x) = -x.

Evaluate g(-1) using the expression g(x) = -x. Substitute -1 for x, which gives g(-1) = -(-1).

Simplify the expression -(-1) to find the value of g(-1).

Now, use the result from g(-1) as the new input for the function g(x) again. Determine which expression of g(x) applies to this new input value.

Evaluate g(g(-1)) using the appropriate expression for g(x) based on the new input value obtained from the previous step.

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Key Concepts

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

Composition of Functions

Composition of functions involves combining two functions where the output of one function becomes the input of another. For example, if you have functions f(x) and g(x), the composition g(f(x)) means you first apply f to x, then apply g to the result. Understanding this concept is crucial for evaluating expressions like g(g(-1)).

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, g(x) has two different rules depending on the value of x: one for -2 ≤ x < 0 and another for 0 ≤ x ≤ 2. Recognizing which part of the function to use based on the input is essential for correctly evaluating g(-1) and subsequently g(g(-1)).

Function Evaluation

Function evaluation is the process of finding the output of a function for a given input. This involves substituting the input value into the function's formula. For instance, to evaluate g(-1), you need to determine which piece of the piecewise function applies and then compute the result accordingly, which is a key step in solving the problem.

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Related Practice

Textbook Question

The accompanying figure shows the graph of a function f(x) with domain [0,2] and range [0,1]. Find the domains and ranges of the following functions, and sketch their graphs.

<IMAGE>

c. 2f(x)

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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

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.

c. y = x¹/⁴

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Textbook Question

Functions

In Exercises 7 and 8, which of the graphs are graphs of functions of x, and which are not? Give reasons for your answers.

b. <IMAGE>

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Textbook Question

Find the largest interval on which the given function is increasing.

c. g(x) = (3x - 1)¹/³