1. **State the problem:** We are given two functions: $$f(x) = 2x - 3$$ $$g(x) = x^4 - 2x^3 + x$$ We need to find the sum and difference of these functions, i.e., $(f + g)(x)$ and $(f - g)(x)$, and state their domains. 2. **Recall the formulas:** $$(f + g)(x) = f(x) + g(x)$$ $$(f - g)(x) = f(x) - g(x)$$ 3. **Calculate $(f + g)(x)$:** $$egin{aligned} (f + g)(x) &= (2x - 3) + (x^4 - 2x^3 + x) \\ &= x^4 - 2x^3 + 2x + x - 3 \\ &= x^4 - 2x^3 + 3x - 3 \end{aligned}$$ 4. **Calculate $(f - g)(x)$:** $$egin{aligned} (f - g)(x) &= (2x - 3) - (x^4 - 2x^3 + x) \\ &= 2x - 3 - x^4 + 2x^3 - x \\ &= -x^4 + 2x^3 + (2x - x) - 3 \\ &= -x^4 + 2x^3 + x - 3 \end{aligned}$$ 5. **State the domain:** Both $f(x)$ and $g(x)$ are polynomials, and polynomials are defined for all real numbers. Therefore: - Domain of $(f + g)(x)$ is all real numbers, i.e., $(-\infty, \infty)$. - Domain of $(f - g)(x)$ is all real numbers, i.e., $(-\infty, \infty)$. **Final answers:** $$(f + g)(x) = x^4 - 2x^3 + 3x - 3$$ $$(f - g)(x) = -x^4 + 2x^3 + x - 3$$ Domain of $(f + g)(x)$: $(-\infty, \infty)$ Domain of $(f - g)(x)$: $(-\infty, \infty)$