**Problem Statement** For \( f(x) = 5 \) and \( g(x) = 3x^2 + 2 \), find the domains and ranges of \( f \), \( g \), and the functions \( \frac{f}{g} \) and \( \frac{g}{f} \). --- **Solution Explanation** 1. **Function \( f(x) = 5 \)** - **Domain of \( f \)**: The function \( f(x) = 5 \) is a constant function. Its domain is all real numbers, denoted as \( (-\infty, \infty) \). - **Range of \( f \)**: Since \( f(x) \) is constant, its range is \( \{5\} \). 2. **Function \( g(x) = 3x^2 + 2 \)** - **Domain of \( g \)**: The function \( g(x) = 3x^2 + 2 \) is a polynomial, which has a domain of all real numbers, \( (-\infty, \infty) \). - **Range of \( g \)**: Since \( g(x) \) is a parabola opening upwards with vertex at its minimum point, the range is \([2, \infty)\). 3. **Function \( \frac{f}{g} = \frac{5}{3x^2 + 2} \)** - **Domain of \( \frac{f}{g} \)**: The denominator \( g(x) = 3x^2 + 2 \) is never zero for real numbers. Therefore, the domain is \( (-\infty, \infty) \). - **Range of \( \frac{f}{g} \)**: Since \( g(x) \geq 2 \), the expression \( \frac{5}{3x^2 + 2} \) will attain values from \((0, \frac{5}{2}]\). 4. **Function \( \frac{g}{f} = \frac{3x^2 + 2}{5} \)** - **Domain of \( \frac{g}{f} \)**: The domain is all real numbers, similar to \( g(x) \), since the denominator \( f(x) \) is simply

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