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) = \boxed{} (do not evaluate.)

Explanation:

To find \((f \circ g)(x)\), we use the definition of function composition, which states that \((f \circ g)(x)=f(g(x))\). This means we substitute \(g(x)\) into the function \(f\) wherever there is an \(x\) in \(f\).

Step 1: Recall the functions

We are given that \(f(x) = x^3\) and \(g(x)=x^2 + 7x-4\).

Step 2: Substitute \(g(x)\) into \(f\)

To find \(f(g(x))\), we replace the \(x\) in \(f(x)=x^3\) with \(g(x)=x^2 + 7x - 4\). So we have:
\(f(g(x))=(g(x))^3\)
Substituting \(g(x)=x^2 + 7x - 4\) into the above expression, we get:
\(f(g(x))=(x^2 + 7x - 4)^3\)

Answer:

\(\boxed{(x^2 + 7x - 4)^3}\)