For \( f(x)=\sqrt{x} \) and \( g(x)=6 x+7 \), find the following composite functions and state the domain of each. \( \begin{array}{lll}\text { (a) } f \circ g & \text { (b) } g \circ f & \text { (c) } f \circ f\end{array} \quad \) (d) \( g \circ g \) \( \begin{array}{ll}\text { (a) }(f \circ g)(x)=\sqrt{6 x+7} \\ \text { (Simplify your answer.) } \\ \text { The domain of } f \circ g \text { is } x \geq-\frac{7}{6} & \text { (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Simplify } \\ \text { your answer.) }\end{array} \)

To find the composite functions, we can follow these steps: (a) For \( f \circ g \): \[ (f \circ g)(x) = f(g(x)) = f(6x + 7) = \sqrt{6x + 7} \] The domain of \( f(g(x)) \) is determined by the expression inside the square root needing to be non-negative: \[ 6x + 7 \geq 0 \implies x \geq -\frac{7}{6} \] Thus, the domain in interval notation is: \( \left[-\frac{7}{6}, \infty\right) \). (b) For \( g \circ f \): \[ (g \circ f)(x) = g(f(x)) = g(\sqrt{x}) = 6\sqrt{x} + 7 \] The domain of \( g(f(x)) \) is determined by \( x \) needing to be non-negative since \( f(x) = \sqrt{x} \): \[ x \geq 0 \] So the domain in interval notation is: \( [0, \infty) \). (c) Now, for \( f \circ f \): \[ (f \circ f)(x) = f(f(x)) = f(\sqrt{x}) = \sqrt{\sqrt{x}} = x^{1/4} \] The domain of \( f(f(x)) \) is also determined by the square root, which requires \( x \geq 0 \): Thus, the domain in interval notation is: \( [0, \infty) \). (d) Lastly, for \( g \circ g \): \[ (g \circ g)(x) = g(g(x)) = g(6x + 7) = 6(6x + 7) + 7 = 36x + 42 + 7 = 36x + 49 \] The domain of \( g(g(x)) \) is defined for all \( x \) in the real numbers since there are no restrictions: Therefore, the domain in interval notation is: \( (-\infty, \infty) \). In summary: - \( (f \circ g)(x) = \sqrt{6x + 7} \) with a domain of \( \left[-\frac{7}{6}, \infty\right) \). - \( (g \circ f)(x) = 6\sqrt{x} + 7 \) with a domain of \( [0, \infty) \). - \( (f \circ f)(x) = x^{1/4} \) with a domain of \( [0, \infty) \). - \( (g \circ g)(x) = 36x + 49 \) with a domain of \( (-\infty, \infty) \).