1. **Problem statement:** Given functions $f(x) = 3x + 5$ and $g(x) = x^2 - 2$, perform the following operations: a) $f(x) + g(x)$ b) $(f - g)(x)$ c) $(g \cdot f)(x)$ d) $f(g(x))$ 2. **Recall function operations:** - Addition: $(f + g)(x) = f(x) + g(x)$ - Subtraction: $(f - g)(x) = f(x) - g(x)$ - Multiplication: $(g \cdot f)(x) = g(x) \times f(x)$ - Composition: $f(g(x))$ means substitute $g(x)$ into $f$ 3. **Calculate each:** a) $f(x) + g(x) = (3x + 5) + (x^2 - 2) = x^2 + 3x + (5 - 2) = x^2 + 3x + 3$ b) $(f - g)(x) = (3x + 5) - (x^2 - 2) = 3x + 5 - x^2 + 2 = -x^2 + 3x + 7$ c) $(g \cdot f)(x) = (x^2 - 2)(3x + 5) = x^2 \times 3x + x^2 \times 5 - 2 \times 3x - 2 \times 5 = 3x^3 + 5x^2 - 6x - 10$ d) $f(g(x)) = f(x^2 - 2) = 3(x^2 - 2) + 5 = 3x^2 - 6 + 5 = 3x^2 - 1$ **Final answers:** a) $x^2 + 3x + 3$ b) $-x^2 + 3x + 7$ c) $3x^3 + 5x^2 - 6x - 10$ d) $3x^2 - 1$