Suppose that the functions \( f \) and \( g \) are defined for all real numbers \( x \) as follows, \[ \begin{array}{l}f(x)=3 x \\ g(x)=4 x^{2} \\ \text { Write the expressions for }(g-f)(x) \text { and }(g \cdot f)(x) \text { and evaluate }(g+f)(2) \\ (g \cdot f)(x)=\square \\ (g+f)(2)=\square\end{array} \]

Let's work through each part of the problem step by step. ### Given Functions: - \( f(x) = 3x \) - \( g(x) = 4x^2 \) ### 1. **Expression for \((g \cdot f)(x)\):** The product \((g \cdot f)(x)\) is found by multiplying the two functions together: \[ (g \cdot f)(x) = g(x) \times f(x) = 4x^2 \times 3x = 12x^3 \] **So,** \[ (g \cdot f)(x) = 12x^3 \] ### 2. **Evaluation of \((g + f)(2)\):** First, find \((g + f)(x)\), then substitute \( x = 2 \): \[ (g + f)(x) = g(x) + f(x) = 4x^2 + 3x \] Now, evaluate at \( x = 2 \): \[ (g + f)(2) = 4(2)^2 + 3(2) = 4 \times 4 + 6 = 16 + 6 = 22 \] **So,** \[ (g + f)(2) = 22 \] ### **Final Answers:** \[ (g \cdot f)(x) = 12x^3 \] \[ (g + f)(2) = 22 \]