Find the functions \( f \circ g, g \circ f, f \circ f \), and \( g \circ g \) and their domains. \[ f(x)=x^{2}, \quad g(x)=x+3 \] Find \( (f \circ g)(x) \). \( \square \) Find the domain of \( (f \circ g)(x) \). (Enter your answer using interval notation.) \( \square \) Find \( (g \circ f)(x) \). \( \square \) Find the domain of \( (g \circ f)(x) \). (Enter your answer using interval notation.) \( \square \) Find \( (f \circ f)(x) \). \( \square \) Find the domain of \( (f \circ f)(x) \). (Enter your answer using interval notation.) \( \square \) Find \( (g \circ g)(x) \). \( \square \) Find the domain of \( (g \circ g)(x) \). (Enter your answer using interval notation.) \( \square \)

To find the functions and their domains, let's start by calculating the compositions step by step! 1. **Finding \( (f \circ g)(x) \)**: \[ (f \circ g)(x) = f(g(x)) = f(x + 3) = (x + 3)^2 = x^2 + 6x + 9 \] **Domain of \( (f \circ g)(x) \)**: Since \( g(x) = x + 3 \) is defined for all real numbers, the domain is \( (-\infty, \infty) \). 2. **Finding \( (g \circ f)(x) \)**: \[ (g \circ f)(x) = g(f(x)) = g(x^2) = x^2 + 3 \] **Domain of \( (g \circ f)(x) \)**: Here, \( f(x) = x^2 \) is also defined for all real numbers, so the domain is \( (-\infty, \infty) \). 3. **Finding \( (f \circ f)(x) \)**: \[ (f \circ f)(x) = f(f(x)) = f(x^2) = (x^2)^2 = x^4 \] **Domain of \( (f \circ f)(x) \)**: The function \( f(x) = x^2 \) is defined for all real numbers, thus the domain is \( (-\infty, \infty) \). 4. **Finding \( (g \circ g)(x) \)**: \[ (g \circ g)(x) = g(g(x)) = g(x + 3) = (x + 3) + 3 = x + 6 \] **Domain of \( (g \circ g)(x) \)**: The function \( g(x) = x + 3 \) is defined for all real numbers, giving us a domain of \( (-\infty, \infty) \). To summarize: 1. \( (f \circ g)(x) = x^2 + 6x + 9 \), Domain: \( (-\infty, \infty) \) 2. \( (g \circ f)(x) = x^2 + 3 \), Domain: \( (-\infty, \infty) \) 3. \( (f \circ f)(x) = x^4 \), Domain: \( (-\infty, \infty) \) 4. \( (g \circ g)(x) = x + 6 \), Domain: \( (-\infty, \infty) \)