Question

given functions f and g, find (a) (f ∘ g)(x) and its domain, and (b) (g ∘ f)(x) and its domain.

f(x) = x³, g(x) = x² + 7x - 3

(a) (f ∘ g)(x) =
(do not evaluate.)
the domain of (f ∘ g)(x) is
(type your answer in interval notation.)

(b) (g ∘ f)(x) =
(do not factor.)

Explanation:

Step1: Recall function composition

To find \((g \circ f)(x)\), we substitute \(f(x)\) into \(g(x)\). So \(g(f(x)) = g(x^3)\).

Step2: Substitute into \(g\)

Since \(g(x)=x^2 + 7x - 3\), replacing \(x\) with \(x^3\) gives \(g(x^3)=(x^3)^2+7(x^3)-3\).

Step3: Simplify the expression

Simplify \((x^3)^2\) using the power - of - a - power rule \((a^m)^n=a^{mn}\), so \((x^3)^2 = x^{6}\). Then the expression becomes \(x^{6}+7x^{3}-3\).

Answer:

\(x^{6}+7x^{3}-3\)