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.)

To find \( (g \circ f)(x) \), we need to substitute \( f(x) \) into \( g(x) \). We have: \[ f(x) = 5x - 8 \] So, \[ g(f(x)) = g(5x - 8) = (5x - 8)^2 \] Expanding this, we get: \[ (5x - 8)^2 = 25x^2 - 80x + 64 \] Thus, \( (g \circ f)(x) = 25x^2 - 80x + 64 \). Now for the domain of \( g \circ f \): The function \( g \) is defined for all real numbers since \( g(x) = x^2 \), and since \( f \) is also defined for all real numbers, the composition \( g(f(x)) \) is also defined for all \( x \in (-\infty, \infty) \). Putting this all together, we have: \[ (b) \quad (g \circ f)(x) = 25x^2 - 80x + 64 \] The domain of \( g \circ f \) is \( (-\infty, \infty) \).