Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경

Briggs 3rd Edition

Ch. 1 - Functions

문제 19c

1장, 문제 19c

Composite functions
Let ƒ(x) = x³, g (x) = sin x and h(x) = √x.
Find ƒ(g(h( x))).

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Identify the innermost function in the composite function $ f(g(h(x))) $. Here, it is $ h(x) = \sqrt{x} $.

Substitute $ h(x) $ into $ g(x) $, resulting in $ g(h(x)) = g(\sqrt{x}) = \sin(\sqrt{x}) $.

Now, substitute $ g(h(x)) $ into $ f(x) $, resulting in $ f(g(h(x))) = f(\sin(\sqrt{x})) $.

Since $ f(x) = x^3 $, replace $ x $ with $ \sin(\sqrt{x}) $ to get $ (\sin(\sqrt{x}))^3 $.

Thus, the expression for $ f(g(h(x))) $ is $ (\sin(\sqrt{x}))^3 $.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Composite Functions

Composite functions are formed when one function is applied to the result of another function. In mathematical notation, if you have two functions f(x) and g(x), the composite function f(g(x)) means you first apply g to x, and then apply f to the result of g. Understanding how to combine functions is essential for solving problems involving multiple functions.

Function Notation

Function notation is a way to represent functions and their operations clearly. For example, f(x) denotes a function f evaluated at x. This notation helps in understanding how to manipulate and combine functions, especially when dealing with compositions like f(g(h(x))). Recognizing how to read and interpret these notations is crucial for solving calculus problems.

Basic Functions

Basic functions such as polynomial functions, trigonometric functions, and root functions form the foundation of calculus. In this question, f(x) = x³ is a polynomial function, g(x) = sin x is a trigonometric function, and h(x) = √x is a root function. Knowing the properties and behaviors of these functions is vital for accurately performing operations like composition.

관련 실천

교과서 질문

Composite functions

Let ƒ(x) = x³, g (x) = sin x and h(x) = √x .

Evaluate h(g( π/2)).

356

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교과서 질문

Demand function Sales records indicate that if Blu-ray players are priced at \$250, then a large store sells an average of 12 units per day. If they are priced at \$200, then the store sells an average of 15 units per day. Find and graph the linear demand function for Blu-ray sales. For what prices is the demand function defined?

359

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교과서 질문

Composite functions

Let ƒ(x) = x³, g (x) = sin x and h(x) = √x.

Find the domain of g o ƒ.

355

views

교과서 질문

Composite functions

Let ƒ(x) = x³, g (x) = sin x and h(x) = √x .

Find h (ƒ (x)).

350

views

교과서 질문

Yeast growth Consider a colony of yeast cells that has the shape of a cylinder. As the number of yeast cells increases, the cross-sectional area A (in mm²) of the colony increases but the height of the colony remains constant. If the colony starts from a single cell, the number of yeast cells (in millions) is approximated by the linear function N(A) - CₛA, where the constant Cₛ is known as the cell-surface coefficient. Use the given information to determine the cell-surface coefficient for each of the following colonies of yeast cells, and find the number of yeast cells in the colony when the cross-sectional area A reaches 150 mm². (Source: Letters in Applied Microbiology, 594, 59, 2014)

The scientific name of baker’s or brewer’s yeast (used in making bread, wine, and beer) is Saccharomyces cerevisiae. When the cross-sectional area of a colony of this yeast reaches 100 mm², there are 571 million yeast cells.

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Composite functionsLet ƒ(x) = x³, g (x) = sin x and h(x) = √x.Find ƒ(g(h( x))).

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Composite Functions

Function Notation

Basic Functions

Composite functions
Let ƒ(x) = x³, g (x) = sin x and h(x) = √x.
Find ƒ(g(h( x))).