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

Verified step by step guidance

1. The composition of functions \( g \circ f \) means applying function \( g \) to the result of function \( f \). In this case, \( g \circ f(x) = g(f(x)) = g(x^3) = \sin(x^3).\)
2. The function \( f(x) = x^3 \) is a polynomial, and polynomials are defined for all real numbers. Therefore, the domain of \( f(x) \) is all real numbers, \( (-\infty, \infty) \).
3. The sine function, \( \sin x \), is defined for all real numbers. Therefore, the domain of \( g(x) \) is also all real numbers, \( (-\infty, \infty) \).
4. Since \( f(x) = x^3 \) is defined for all real numbers and \( g(x) = \sin x \) is also defined for all real numbers, the composition \( g \circ f(x) = \sin(x^3) \) is defined for all real numbers.
5. The domain of \( g \circ f \) is the set of all real numbers, \( (-\infty, \infty) \), because both \( f(x) \) and \( g(x) \) are defined for all real numbers.

Key Concepts

Composite Functions

A composite function is formed when one function is applied to the result of another function. In this case, g o ƒ means we first apply the function ƒ to x, and then apply g to the result of ƒ(x). Understanding how to combine functions is essential for determining the overall behavior and properties of the composite function.

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. To find the domain of g o ƒ, we must ensure that the output of ƒ(x) falls within the domain of g(x).

Function Behavior and Restrictions

Different functions have specific restrictions that affect their domains. For example, the function g(x) = sin x is defined for all real numbers, while ƒ(x) = x³ is also defined for all real numbers. However, if we were to consider a function like h(x) = √x, it would impose restrictions since it is only defined for x ≥ 0. Understanding these behaviors is crucial for determining the valid inputs for composite functions.