Textbook Question
Composite functions
Let ƒ(x) = x³, g (x) = sin x and h(x) = √x .
Evaluate h(g( π/2)).
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Ch. 1 - Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 1, Problem 19c
Composite functions
Let ƒ(x) = x³, g (x) = sin x and h(x) = √x.
Find ƒ(g(h( x))).
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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Evaluate Composite Functions - Special Cases
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.
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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.
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Related Practice
Textbook Question
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?
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Textbook Question
Composite functions
Let ƒ(x) = x³, g (x) = sin x and h(x) = √x.
Find the domain of g o ƒ.
355
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Textbook Question
Composite functions
Let ƒ(x) = x³, g (x) = sin x and h(x) = √x .
Find h (ƒ (x)).
350
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Textbook Question
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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