Given functions f and g, find (a)(ƒ∘g)(x)(ƒ∘g)(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 78a

3장, 문제 78a

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

검증된 단계별 안내

Identify the given functions: $f(x) = x + 2$ and $g(x) = x^4 + x^2 - 4$.

To find the composition $(f \circ g)(x)$, substitute $g(x)$ into $f(x)$, which means replacing every $x$ in $f(x)$ with $g(x)$: $(f \circ g)(x) = f(g(x)) = g(x) + 2$.

Write the composed function explicitly by plugging in $g(x)$: $(f \circ g)(x) = (x^4 + x^2 - 4) + 2$.

Simplify the expression inside the composition by combining like terms: $(f \circ g)(x) = x^4 + x^2 - 2$.

Determine the domain of $(f \circ g)(x)$ by considering the domain of $g(x)$ first, then ensuring the output of $g(x)$ fits into the domain of $f$. Since both $f$ and $g$ are polynomials, their domains are all real numbers, so the domain of $(f \circ g)(x)$ is all real numbers.

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주요 개념

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Function Composition

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

Polynomial Functions

Both f(x) = x + 2 and g(x) = x^4 + x^2 - 4 are polynomial functions, which are expressions involving variables raised to whole-number powers with coefficients. Recognizing their polynomial nature helps in simplifying and evaluating compositions without domain restrictions from radicals or denominators.

Domain of Composite Functions

The domain of (f∘g)(x) consists of all x-values in the domain of g for which g(x) is in the domain of f. Since f and g are polynomials, their domains are all real numbers, but verifying this ensures the composite function is defined for all inputs.

Given functions f and g, find (a)(ƒ∘g)(x)(ƒ∘g)(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