Given functions f and g, find (b)$(g\circ \text{ƒ})\left(x\right)$ and its domain. See Examples 6 and 7.
$\text{ƒ}\left(x\right)=x+2,g\left(x\right)=x^4+x^2-4$

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 evaluated at the output of f(x).
2. Start by substituting f(x) into g(x). Since f(x) = x + 2, replace every x in g(x) with (x + 2). So, write g(f(x)) as g(x + 2) = (x + 2)^4 + (x + 2)^2 - 4.
3. Next, expand the expressions (x + 2)^4 and (x + 2)^2 using binomial expansion or repeated multiplication to simplify g(f(x)) into a polynomial in terms of x.
4. After expanding, combine like terms to write the composition (g \circ f)(x) as a simplified polynomial expression.
5. To find the domain of (g \circ f)(x), consider the domain of f(x) and the domain of g(x). Since f(x) = x + 2 is defined for all real numbers, and g(x) = x^4 + x^2 - 4 is also defined for all real numbers, 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)). This means you first evaluate f(x), then use that output as the input for g. Understanding this process is essential to correctly find (g∘f)(x).

Evaluating Polynomial Functions

Evaluating polynomial functions requires substituting the input value into the polynomial expression and simplifying. For example, given g(x) = x^4 + x^2 - 4, you replace x with the input and calculate the powers and sums accordingly. This skill is necessary to compute g(f(x)).

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. Since both f and g are polynomials with domain all real numbers, the domain of (g∘f)(x) is all real numbers, but understanding this concept is crucial for more complex cases.