Given functions f and g, (b)$(g\circ \text{ƒ})\left(x\right)$ and its domain. See Examples 6 and 7.
$\text{ƒ}\left(x\right)=x^3,g\left(x\right)=x^2+3x-1$

Verified step by step guidance

1. Recall that the composition of functions (g \circ f)(x) means g(f(x)), which is the function g applied to the output of f(x).
2. Given f(x) = x^3, substitute f(x) into g(x) to find (g \circ f)(x). This means replace every x in g(x) with f(x) = x^3.
3. Write the expression for (g \circ f)(x) as g(f(x)) = (f(x))^2 + 3(f(x)) - 1, which becomes (x^3)^2 + 3(x^3) - 1.
4. Simplify the expression by applying the exponent and multiplication: (x^3)^2 = x^{6}, so the expression becomes x^{6} + 3x^{3} - 1.
5. Determine the domain of (g \circ f)(x) by considering the domain of f(x) and g(x). Since both f(x) = x^3 and g(x) = x^2 + 3x - 1 are polynomials, their domains are all real numbers, so the domain of (g \circ f)(x) is all real numbers.

Key Concepts

Function Composition

Function composition involves applying one function to the result of another, denoted as (g∘f)(x) = g(f(x)). It means you first evaluate f(x), then use that output as the input for g. Understanding this process is essential to correctly form the composite function.

Evaluating Polynomial Functions

Evaluating polynomial functions requires substituting the input value into the polynomial expression and simplifying. For example, f(x) = x^3 means cubing the input x. This skill is necessary to find g(f(x)) by substituting f(x) into g.

Domain of Composite Functions

The domain of a composite function (g∘f)(x) consists of all x-values in the domain of f for which f(x) is in the domain of g. Identifying this domain ensures the composite function is defined and helps avoid invalid inputs.