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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 1.19e
Chapter 1, Problem 1.19e

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

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1
Determine the domain of g(x) = \(\sin\) x. Since \(\sin\) x is defined for all real numbers, the domain of g(x) is all real numbers.
Consider the function f(x) = x^3. The function f(x) is a polynomial and is defined for all real numbers, so its domain is all real numbers.
To find the domain of the composite function (f \(\circ\) g)(x) = f(g(x)), substitute g(x) into f(x) to get f(\(\sin\) x) = (\(\sin\) x)^3.
Since \(\sin\) x is defined for all real numbers and f(x) = x^3 is defined for all real numbers, the composite function f(\(\sin\) x) is also defined for all real numbers.
Therefore, the domain of the composite function f \(\circ\) g is all real numbers.

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Key Concepts

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

Composite Functions

A composite function is formed when one function is applied to the result of another function. In mathematical terms, if you have two functions f(x) and g(x), the composite function f(g(x)) is denoted as f o g. Understanding how to combine functions is essential for analyzing their behavior and determining their domains.
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Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For composite functions, the domain is influenced by both the inner function and the outer function. It is crucial to identify any restrictions, such as values that would lead to undefined expressions, to accurately determine the overall domain of the composite function.
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Trigonometric Functions and Their Domains

Trigonometric functions, such as sine, have specific domains and ranges. The sine function, g(x) = sin(x), is defined for all real numbers, but when combined with other functions, its output must also fit within the domain of the outer function. In this case, since g(x) is the inner function for f(g(x)), understanding its output is vital for finding the domain of the composite function f o g.
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Related Practice
Textbook Question

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What is the population 4 days4~\(\text{days}\) after the first observation?

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Textbook Question

What are the three Pythagorean identities for the trigonometric functions?

Textbook Question

A culture of bacteria has a population of 150150 cells when it is first observed. The population doubles every 12 hr12~\(\text{hr}\), which means its population is governed by the function p(t)=150⋅2t12p\(\left\)(t\(\right\))=150\(\cdot{2^{\frac{t}{12}\)}}, where tt is the number of hours after the first observation.

How long does it take the population to triple in size?

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