1. Problem: Find $f(g(x))$ given piecewise functions $f(g(x)) = \begin{cases} -3x + 10, & x < 3 \\ 3x - 8, & x \geq 3 \end{cases}$. Explanation: This is a piecewise function defined by two linear expressions depending on $x$. 2. Problem: Find $f(x)$ given $f(x) = \begin{cases} 2x - 7, & x \leq 2 \\ 5 - 3x, & x > 2 \end{cases}$. Explanation: Another piecewise function with two linear parts. 3. Problem: Simplify $f(x) = \frac{x + 28}{3}$. Explanation: This is a linear function divided by 3. 4. Problem: Simplify $f(x) = \frac{2x^2 - 14}{x + 1996}$. Explanation: Factor numerator: $2x^2 - 14 = 2(x^2 - 7)$. 5. Problem: Simplify $f(x) = \frac{2x + 5}{3}$. Explanation: Linear function divided by 3. 6. Problem: Simplify quadratic polynomial $2x^2 + 5x - 2$. Explanation: Can factor or leave as is. 7. Problem: Simplify $\frac{2x - 3}{5}$ and $\frac{x^2 - 4x}{x + 2}$. Explanation: First is linear over constant; second can factor numerator: $x(x - 4)$. 8. Problem: Simplify $f(g(x)) = \frac{2x + 3}{x - 5}$. 9. Problem: Simplify $f(x) = \frac{x + 3}{2x}$. 10. Problem: Simplify $f(x) = \frac{x - 3}{2}$. 11. Problem: Given $\sqrt{x} - 1$, find which expression equals $f(x)$ among options: A) $3|x-1|$ B) $|x| - 3$ C) $3|x| - 1$ D) $|x| - 3|$ Solution steps: 1. For $f(g(x))$ piecewise, the function is already given; no further simplification needed. 2. For $f(x)$ piecewise, similarly, function is defined piecewise. 3. Simplify $\frac{x + 28}{3}$ as is; no further simplification. 4. Factor numerator of $\frac{2x^2 - 14}{x + 1996}$: $$2x^2 - 14 = 2(x^2 - 7)$$ No common factors with denominator, so expression stays as: $$\frac{2(x^2 - 7)}{x + 1996}$$ 5. $\frac{2x + 5}{3}$ is linear over constant; no simplification. 6. Quadratic $2x^2 + 5x - 2$ factorization: Find factors of $2 \times (-2) = -4$ that sum to 5: 4 and -1. Rewrite: $$2x^2 + 5x - 2 = 2x^2 + 4x - x - 2 = 2x(x + 2) -1(x + 2) = (2x - 1)(x + 2)$$ 7. Simplify $\frac{2x - 3}{5}$ as is. Simplify $\frac{x^2 - 4x}{x + 2}$: Factor numerator: $$x^2 - 4x = x(x - 4)$$ No common factor with denominator $x + 2$, so expression stays. 8. $f(g(x)) = \frac{2x + 3}{x - 5}$ is already simplified. 9. $f(x) = \frac{x + 3}{2x}$ is simplified. 10. $f(x) = \frac{x - 3}{2}$ is simplified. 11. For $\sqrt{x} - 1$, check options: - $3|x-1|$ is not equal to $\sqrt{x} - 1$. - $|x| - 3$ is not equal. - $3|x| - 1$ is not equal. - $|x| - 3|$ is not a valid expression. Hence, none match exactly; likely a trick question or typo. Final answers: - Piecewise functions as given. - Factorization: $2x^2 + 5x - 2 = (2x - 1)(x + 2)$. - Simplifications as above.