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 - 4

(a) (f ∘ g)(x) = (x² + 7x - 4)³ (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 Composition of Functions

To find \((g \circ f)(x)\), we substitute \(f(x)\) into \(g(x)\). The formula for composition is \((g \circ f)(x)=g(f(x))\).

Step2: Substitute \(f(x)\) into \(g(x)\)

Given \(f(x) = x^3\) and \(g(x)=x^2 + 7x - 4\), we replace \(x\) in \(g(x)\) with \(f(x)=x^3\). So we get \(g(f(x))=(x^3)^2+7(x^3)-4\).

Step3: Simplify the Expression

Simplify \((x^3)^2\) using the power - of - a - power rule \((a^m)^n=a^{mn}\). Here, \(a = x\), \(m = 3\), \(n = 2\), so \((x^3)^2=x^{6}\). And \(7(x^3)=7x^3\). So \(g(f(x))=x^{6}+7x^{3}-4\).

Answer:

\(x^{6}+7x^{3}-4\)