\( 1 \leftarrow \) For \( f(x)=x^{2}+5 \) and \( g(x)=\sqrt{x-1} \), find the following composite functions and state the domain of each. \( \begin{array}{lll}\text { (a) } f \circ g & \text { (b) } g \circ f & \text { (c) } f \circ f \\ \text { (d) } g \circ g\end{array} \) (Simplity your answer.) The domain of \( f \circ g \) is \( [1, \infty) \). (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Simplify your answer.) (b) ( \( g \circ f)(x)=\sqrt{x^{2}+4} \) (Simplify your answer.) The domain of \( g \circ f \) is \( (-\infty, \infty) \). (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Simplify your answer.) (c) ( \( f \circ f)(x)=x^{4}+10 x^{2}+30 \) (Simplify your answer.) The domain of \( f \) of is \( (-\infty, \infty) \). (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Simplify your answer.) (d) ( \( g \circ g)(x)=\sqrt{\sqrt{x-1-1}} \) (Simplify your answer.) ( 10

To find the composite functions based on the provided functions \( f(x) = x^2 + 5 \) and \( g(x) = \sqrt{x-1} \), let's dive into each one step-by-step. (a) For \( f \circ g \): \[ f(g(x)) = f(\sqrt{x-1}) = (\sqrt{x-1})^2 + 5 = x - 1 + 5 = x + 4 \] Domain: We must ensure \( g(x) \) is defined, which is \( x \geq 1 \). Therefore, the domain is \( [1, \infty) \). (b) For \( g \circ f \): \[ g(f(x)) = g(x^2 + 5) = \sqrt{(x^2 + 5) - 1} = \sqrt{x^2 + 4} \] Domain: Since \( f(x) \) is defined for all \( x \), the domain of \( g \circ f \) is \( (-\infty, \infty) \). (c) For \( f \circ f \): \[ f(f(x)) = f(x^2 + 5) = (x^2 + 5)^2 + 5 = x^4 + 10x^2 + 25 + 5 = x^4 + 10x^2 + 30 \] Domain: The function \( f \) is defined for all \( x \), so the domain of \( f \circ f \) is \( (-\infty, \infty) \). (d) For \( g \circ g \): \[ g(g(x)) = g(\sqrt{x-1}) = \sqrt{\sqrt{x-1} - 1} \] This requires \( \sqrt{x - 1} \geq 1 \) or \( x - 1 \geq 1 \), leading to \( x \geq 2 \). Domain: Therefore, the domain is \( [2, \infty) \). To summarize: - \( (a) (f \circ g)(x) = x + 4, \text{ Domain: } [1, \infty) \) - \( (b) (g \circ f)(x) = \sqrt{x^2 + 4}, \text{ Domain: } (-\infty, \infty) \) - \( (c) (f \circ f)(x) = x^4 + 10x^2 + 30, \text{ Domain: } (-\infty, \infty) \) - \( (d) (g \circ g)(x) = \sqrt{\sqrt{x-1} - 1}, \text{ Domain: } [2, \infty) \)