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) = x⁶ + 7x³ - 4 (do not factor.)
the domain of (g ∘ f)(x) is □.
(type your answer in interval notation.)

Explanation:

Step1: Analyze \( g \circ f(x) \)

\( (g \circ f)(x)=g(f(x)) \). Given \( f(x)=x^3 \), substitute into \( g(x)=x^2 + 7x - 4 \), we get \( g(x^3)=(x^3)^2 + 7(x^3)-4=x^6 + 7x^3 - 4 \), which is a polynomial.

Step2: Determine the domain of a polynomial

Polynomial functions (like \( x^6 + 7x^3 - 4 \)) are defined for all real numbers. So the domain is all real numbers, which in interval notation is \( (-\infty, \infty) \).

Answer:

\( (-\infty, \infty) \)