In Exercises \( 35-42 \), find (a) \( f \) eg and (b) \( g \circ f \). Find the domain of each function and each composite function. \( \begin{array}{ll}\text { 35. } f(x)=\sqrt{x+4}, & g(x)=x^{2} \\ \text { 36. } f(x)=\sqrt[3]{x-5}, & g(x)=x^{3}+1 \\ \text { 37. } f(x)=x^{2}+1, & g(x)=\sqrt{x} \\ \text { 38. } f(x)=x^{2 / 3}, & g(x)=x^{6}\end{array} \)

Let's solve each exercise step by step. ### Exercise 35 **Functions:** - \( f(x) = \sqrt{x + 4} \) - \( g(x) = x^2 \) **(a) Domain of \( f \):** The expression under the square root must be non-negative: \[ x + 4 \geq 0 \implies x \geq -4 \] Thus, the domain of \( f \) is \( [-4, \infty) \). **(b) Composite function \( g \circ f \):** \[ g(f(x)) = g(\sqrt{x + 4}) = (\sqrt{x + 4})^2 = x + 4 \] **Domain of \( g \circ f \):** Since \( f(x) \) has a domain of \( [-4, \infty) \) and \( g(x) \) is defined for all real numbers, the domain of \( g \circ f \) is also \( [-4, \infty) \). ### Exercise 36 **Functions:** - \( f(x) = \sqrt[3]{x - 5} \) - \( g(x) = x^3 + 1 \) **(a) Domain of \( f \):** The cube root function is defined for all real numbers, so: \[ \text{Domain of } f: (-\infty, \infty) \] **(b) Composite function \( g \circ f \):** \[ g(f(x)) = g(\sqrt[3]{x - 5}) = (\sqrt[3]{x - 5})^3 + 1 = x - 5 + 1 = x - 4 \] **Domain of \( g \circ f \):** Since both \( f(x) \) and \( g(x) \) are defined for all real numbers, the domain of \( g \circ f \) is also \( (-\infty, \infty) \). ### Exercise 37 **Functions:** - \( f(x) = x^2 + 1 \) - \( g(x) = \sqrt{x} \) **(a) Domain of \( f \):** The function \( f(x) \) is a polynomial and is defined for all real numbers: \[ \text{Domain of } f: (-\infty, \infty) \] **(b) Composite function \( g \circ f \):** \[ g(f(x)) = g(x^2 + 1) = \sqrt{x^2 + 1} \] **Domain of \( g \circ f \):** The square root function requires the argument to be non-negative: \[ x^2 + 1 \geq 0 \quad \text{(always true for all } x\text{)} \] Thus, the domain of \( g \circ f \) is \( (-\infty, \infty) \). ### Exercise 38 **Functions:** - \( f(x) = x^{2/3} \) - \( g(x) = x^6 \) **(a) Domain of \( f \):** The function \( f(x) \) is defined for all real numbers: \[ \text{Domain of } f: (-\infty, \infty) \] **(b) Composite function \( g \circ f \):** \[ g(f(x)) = g(x^{2/3}) = (x^{2/3})^6 = x^4 \] **Domain of \( g \circ f \):** Since both \( f(x) \) and \( g(x) \) are defined for all real numbers, the domain of \( g \circ f \) is also \( (-\infty, \infty) \). ### Summary of Results - **Exercise 35:** - \( f \): Domain \( [-4, \infty) \) - \( g \circ f \): Domain \( [-4, \infty) \) - **Exercise 36:** - \( f \): Domain \( (-\infty, \infty) \) - \( g \circ f \): Domain \( (-\infty, \infty) \) - **Exercise 37:** - \( f \): Domain \( (-\infty, \infty) \) - \( g \circ f \): Domain \( (-\infty, \infty) \) - **Exercise 38:** - \( f \): Domain \( (-\infty, \infty) \) - \( g \circ f \):