1. **State the problem:** We have two functions: $$f(x) = 6x - 4x^2 - 7x^3$$ $$g(x) = 9x^2 - 5x$$ We need to find: - $$(f + g)(x)$$ and $$(f - g)(x)$$ - Evaluate $$(f + g)(-1)$$ and $$(f - g)(-1)$$ - State the domain of $$(f + g)(x)$$ and $$(f - g)(x)$$. 2. **Find $$(f + g)(x)$$:** $$(f + g)(x) = f(x) + g(x) = (6x - 4x^2 - 7x^3) + (9x^2 - 5x)$$ Combine like terms: $$= 6x - 5x - 4x^2 + 9x^2 - 7x^3$$ $$= (6x - 5x) + (-4x^2 + 9x^2) - 7x^3$$ $$= x + 5x^2 - 7x^3$$ 3. **Find $$(f - g)(x)$$:** $$(f - g)(x) = f(x) - g(x) = (6x - 4x^2 - 7x^3) - (9x^2 - 5x)$$ Distribute the minus: $$= 6x - 4x^2 - 7x^3 - 9x^2 + 5x$$ Combine like terms: $$= (6x + 5x) + (-4x^2 - 9x^2) - 7x^3$$ $$= 11x - 13x^2 - 7x^3$$ 4. **Evaluate $$(f + g)(-1)$$:** Substitute $x = -1$ into $$(f + g)(x) = x + 5x^2 - 7x^3$$ $$= (-1) + 5(-1)^2 - 7(-1)^3$$ Calculate powers: $$= -1 + 5(1) - 7(-1)$$ $$= -1 + 5 + 7$$ $$= 11$$ 5. **Evaluate $$(f - g)(-1)$$:** Substitute $x = -1$ into $$(f - g)(x) = 11x - 13x^2 - 7x^3$$ $$= 11(-1) - 13(-1)^2 - 7(-1)^3$$ Calculate powers: $$= -11 - 13(1) - 7(-1)$$ $$= -11 - 13 + 7$$ $$= -17$$ 6. **State the domain:** Both $f(x)$ and $g(x)$ are polynomials, and polynomials are defined for all real numbers. Therefore, the domain of $$(f + g)(x)$$ and $$(f - g)(x)$$ is all real numbers. **Final answers:** $$(f + g)(x) = x + 5x^2 - 7x^3$$ $$(f + g)(-1) = 11$$ $$(f - g)(x) = 11x - 13x^2 - 7x^3$$ $$(f - g)(-1) = -17$$ The domain of $$(f + g)(x)$$ is all real numbers. The domain of $$(f - g)(x)$$ is all real numbers.