1. **State the problem:** We are given two functions: $$f(x) = 3x^2 - 4x$$ $$g(x) = 9x^2 - 24x + 16$$ We need to perform various operations on these functions and simplify the results. 2. **Formulas and rules:** - Addition: $(f+g)(x) = f(x) + g(x)$ - Subtraction: $(f-g)(x) = f(x) - g(x)$ - Multiplication: $(f \cdot g)(x) = f(x) \times g(x)$ - Division: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$ - Composition: $(g \circ f)(x) = g(f(x))$, $(f \circ f)(x) = f(f(x))$ 3. **Calculate each operation:** **1) $f(x) + g(x)$:** $$f(x) + g(x) = (3x^2 - 4x) + (9x^2 - 24x + 16)$$ Combine like terms: $$= 3x^2 + 9x^2 - 4x - 24x + 16 = 12x^2 - 28x + 16$$ **2) $f(x) - g(x)$:** $$f(x) - g(x) = (3x^2 - 4x) - (9x^2 - 24x + 16)$$ Distribute the minus: $$= 3x^2 - 4x - 9x^2 + 24x - 16 = -6x^2 + 20x - 16$$ **3) $f(x) \cdot g(x)$:** Multiply the polynomials: $$ (3x^2 - 4x)(9x^2 - 24x + 16) $$ Multiply each term: $$= 3x^2 \times 9x^2 + 3x^2 \times (-24x) + 3x^2 \times 16 - 4x \times 9x^2 - 4x \times (-24x) - 4x \times 16$$ Calculate each: $$= 27x^4 - 72x^3 + 48x^2 - 36x^3 + 96x^2 - 64x$$ Combine like terms: $$= 27x^4 - 108x^3 + 144x^2 - 64x$$ **4) $\frac{f(x)}{g(x)}$:** $$\frac{3x^2 - 4x}{9x^2 - 24x + 16}$$ Factor numerator: $$3x^2 - 4x = x(3x - 4)$$ Factor denominator: $$9x^2 - 24x + 16 = (3x - 4)^2$$ Simplify fraction: $$\frac{x(3x - 4)}{(3x - 4)^2} = \frac{x}{3x - 4}, \quad 3x - 4 \neq 0$$ **5) $g(f(x))$:** Substitute $f(x)$ into $g$: $$g(f(x)) = 9(f(x))^2 - 24(f(x)) + 16$$ Calculate $f(x)^2$: $$f(x) = 3x^2 - 4x$$ $$(3x^2 - 4x)^2 = 9x^4 - 24x^3 + 16x^2$$ Now substitute: $$= 9(9x^4 - 24x^3 + 16x^2) - 24(3x^2 - 4x) + 16$$ Expand: $$= 81x^4 - 216x^3 + 144x^2 - 72x^2 + 96x + 16$$ Combine like terms: $$= 81x^4 - 216x^3 + 72x^2 + 96x + 16$$ **6) $g(f(-3))$:** Calculate $f(-3)$: $$f(-3) = 3(-3)^2 - 4(-3) = 3(9) + 12 = 27 + 12 = 39$$ Now calculate $g(39)$: $$g(39) = 9(39)^2 - 24(39) + 16$$ Calculate $39^2 = 1521$: $$= 9(1521) - 936 + 16 = 13689 - 936 + 16 = 12769$$ **7) $f(f(x))$:** Substitute $f(x)$ into $f$: $$f(f(x)) = 3(f(x))^2 - 4(f(x))$$ Recall $f(x) = 3x^2 - 4x$, so: $$(3x^2 - 4x)^2 = 9x^4 - 24x^3 + 16x^2$$ Substitute: $$= 3(9x^4 - 24x^3 + 16x^2) - 4(3x^2 - 4x)$$ Expand: $$= 27x^4 - 72x^3 + 48x^2 - 12x^2 + 16x$$ Combine like terms: $$= 27x^4 - 72x^3 + 36x^2 + 16x$$ **8) $f(f(4))$:** Calculate $f(4)$: $$f(4) = 3(4)^2 - 4(4) = 3(16) - 16 = 48 - 16 = 32$$ Now calculate $f(32)$: $$f(32) = 3(32)^2 - 4(32)$$ Calculate $32^2 = 1024$: $$= 3(1024) - 128 = 3072 - 128 = 2944$$ **Final answers:** 1) $12x^2 - 28x + 16$ 2) $-6x^2 + 20x - 16$ 3) $27x^4 - 108x^3 + 144x^2 - 64x$ 4) $\frac{x}{3x - 4}$ 5) $81x^4 - 216x^3 + 72x^2 + 96x + 16$ 6) $12769$ 7) $27x^4 - 72x^3 + 36x^2 + 16x$ 8) $2944$