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

Verified step by step guidance

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.
2. 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.
3. 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.
4. 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.
5. Therefore, the domain of the composite function f \circ g is all real numbers.

Key Concepts

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.

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.

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.