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.)

Explanation:

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

The function \( (f \circ g)(x)=f(g(x))=(x^{2}+7x - 4)^{3} \). The inner function \( g(x)=x^{2}+7x - 4 \) is a polynomial function. Polynomial functions are defined for all real numbers. The outer function \( f(x)=x^{3} \) is also a polynomial function, defined for all real numbers. When we compose two functions where both are defined for all real numbers, the domain of the composition is all real numbers.

Step2: Determine the domain in interval notation

For a function defined for all real numbers, the interval notation is \( (-\infty, \infty) \).

Answer:

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