Given functions f and g, find (b)(g∘ƒ)(x)(g∘ƒ)(x) and its domain. See Examples 6 and 7.ƒ(x)=x+2,g(x)=x4+x2−4ƒ(x)=x+2, g(x)=$$x^4$+x^2-4$$

Ch. 2 - Graphs and Functions

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Ch. 2 - Graphs and Functions

문제 78b

3장, 문제 78b

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

검증된 단계별 안내

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).

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.

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.

After expanding, combine like terms to write the composition (g $\circ$ f)(x) as a simplified polynomial expression.

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Given functions f and g, find (b)(g∘ƒ)(x)(g∘ƒ)(x) and its domain. See Examples 6 and 7.ƒ(x)=x+2,g(x)=x4+x2−4ƒ(x)=x+2, g(x)=x^4+x^2-4