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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 16j
Chapter 1, Problem 16j

Use the table to evaluate the given compositions. <IMAGE>


ƒ(ƒ(h(3)))

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1
Identify the innermost function in the composition, which is \( h(3) \).
Use the table to find the value of \( h(3) \).
Substitute the value of \( h(3) \) into the next function in the composition, \( f(h(3)) \).
Use the table to find the value of \( f(h(3)) \).
Substitute the value of \( f(h(3)) \) into the outermost function, \( f(f(h(3))) \), and use the table to find the final value.

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Key Concepts

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

Function Composition

Function composition involves applying one function to the result of another function. If you have two functions, f(x) and g(x), the composition f(g(x)) means you first apply g to x, and then apply f to the result. This concept is fundamental in calculus as it allows for the evaluation of complex expressions by breaking them down into simpler parts.
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Evaluate Composite Functions - Special Cases

Evaluating Functions

Evaluating a function means substituting a specific input value into the function to find the output. For example, if f(x) = x + 2, then f(3) = 3 + 2 = 5. In the context of function composition, you will need to evaluate the inner function first before using its output as the input for the outer function.
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Nested Functions

Nested functions occur when a function is applied within another function. In the expression f(f(h(3))), h(3) is evaluated first, then the result is used as the input for f, and finally, that output is again used as the input for f. Understanding how to handle nested functions is crucial for correctly evaluating compositions in calculus.
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Properties of Functions
Related Practice
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Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined.


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Use the table to evaluate the given compositions. <IMAGE>


g(h(ƒ(4)))

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Use the table to evaluate the given compositions. <IMAGE>


h(h(h(0)))

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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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Textbook Question

Use the table to evaluate the given compositions. <IMAGE>


g(ƒ(h(4)))

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