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

Name: Calculus: Early Transcendentals
Author: Briggs
ISBN: 9780136847243

Verified step by step guidance

insert step 1> Identify the functions involved in the composition. Here, we have $ f(x) = x^3 $ and $ g(x) = \sin x $.

insert step 2> Understand the composition $ (f \circ g)(x) = f(g(x)) $. This means we first apply $ g(x) $ and then apply $ f $ to the result.

insert step 3> Determine the domain of $ g(x) = \sin x $. The sine function is defined for all real numbers, so the domain of $ g $ is $ (-\infty, \infty) $.

insert step 4> Consider the domain of $ f(x) = x^3 $. The cube function is also defined for all real numbers, so the domain of $ f $ is $ (-\infty, \infty) $.

insert step 5> Since both $ g(x) $ and $ f(x) $ are defined for all real numbers, the domain of $ f \circ g $ is the same as the domain of $ g(x) $, which is $ (-\infty, \infty) $.

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 notation, if f(x) and g(x) are two functions, the composite function f(g(x)) is evaluated by first calculating g(x) and then applying f to that result. Understanding how to combine functions is essential for solving problems involving their domains and ranges.

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 determined by the inner function and any restrictions imposed by the outer function. Identifying the domain is crucial for ensuring that all operations within the functions are valid and do not lead to undefined expressions.

Restrictions on Functions

Restrictions on functions arise from operations that can limit the input values. For example, the square root function h(x) = √x is only defined for x ≥ 0, while the sine function g(x) = sin x is defined for all real numbers. When finding the domain of composite functions, it is important to consider these restrictions to ensure that the resulting function is valid across its entire domain.

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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.

Textbook Question

Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined.

tan (15π/4)

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Textbook Question

Composite functions

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

Find h (ƒ (x)).