For \( f(x)=5 x-8 \) and \( g(x)=x^{2} \), find the following composite functions and state the domain of each. \( \begin{array}{lll}\text { (a) } f \circ g & \text { (b) } g \circ f & \text { (c) } f \circ f\end{array} \) (d) \( g \circ g \) (a) \( (f \circ g)(x)=5 x^{2}-8 \) (Simplify your answer. Do not factor.) The domain of \( f \circ g \) is \( (-\infty, \infty) \). (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Simplify your answer.) (b) \( g \circ f)(x)=\square \) (Simplify your answer. Do not factor.)

We are given the functions   f(x) = 5x – 8  and  g(x) = x². To find each composite function, we substitute one function into the other as indicated. ───────────────────────────── (a) f ∘ g We have already:   (f ∘ g)(x) = f(g(x)) = f(x²) = 5x² – 8. Since g(x) = x² is defined for all x and f is likewise defined for all real numbers, the domain of f ∘ g is (–∞, ∞). ───────────────────────────── (b) g ∘ f We compute:   (g ∘ f)(x) = g(f(x)) = g(5x – 8) = (5x – 8)². Since f(x) = 5x – 8 is defined for all x and squaring is defined for all numbers, the domain is (–∞, ∞). ───────────────────────────── Thus, the answer for part (b) is:   (g ∘ f)(x) = (5x – 8)², with domain: (–∞, ∞).