1. **Problem:** Graph the equation $y = -3x + 2$, determine domain and range, and analyze function properties. 2. **Formula and rules:** This is a linear function of the form $y = mx + b$ where $m$ is slope and $b$ is y-intercept. 3. **Domain:** For linear functions, domain is all real numbers, $\mathbb{R}$. 4. **Range:** Also all real numbers, $\mathbb{R}$, since line extends infinitely. 5. **Function check:** Each $x$ maps to exactly one $y$, so it is a function. 6. **One-to-one check:** Since slope $m = -3 \neq 0$, function is one-to-one. 7. **Onto check:** The function covers all real $y$ values, so it is onto $\mathbb{R}$. 8. **Discrete or continuous:** The function is continuous because it is defined for all real $x$. 9. **Intermediate work:** None needed for linear function. 10. **Final answer:** Domain: $\mathbb{R}$, Range: $\mathbb{R}$, Function: yes, One-to-one: yes, Onto: yes, Continuous. --- 1. **Problem:** Graph the equation $y = 2x^2$, determine domain and range, and analyze function properties. 2. **Formula and rules:** Quadratic function $y = ax^2$ with $a=2$. 3. **Domain:** All real numbers, $\mathbb{R}$. 4. **Range:** Since $a>0$, parabola opens upward, minimum at $y=0$, so range is $[0, \infty)$. 5. **Function check:** Each $x$ has one $y$, so function. 6. **One-to-one check:** Not one-to-one because $f(x) = f(-x)$. 7. **Onto check:** Not onto $\mathbb{R}$ because $y<0$ values are not reached. 8. **Discrete or continuous:** Continuous over $\mathbb{R}$. 9. **Final answer:** Domain: $\mathbb{R}$, Range: $[0, \infty)$, Function: yes, One-to-one: no, Onto: no, Continuous. --- 1. **Problem:** Evaluate $f(-8)$ for $f(x) = 5x^3 + 1$. 2. **Formula:** $f(x) = 5x^3 + 1$. 3. **Calculation:** $$f(-8) = 5(-8)^3 + 1 = 5(-512) + 1 = -2560 + 1 = -2559$$ 4. **Final answer:** $f(-8) = -2559$. --- 1. **Problem:** Evaluate $f(-5)$ for $f(x) = 3x + 2$. 2. **Calculation:** $$f(-5) = 3(-5) + 2 = -15 + 2 = -13$$ 3. **Final answer:** $f(-5) = -13$. --- 1. **Problem:** Evaluate $f(9)$ for $f(x) = 3x + 2$. 2. **Calculation:** $$f(9) = 3(9) + 2 = 27 + 2 = 29$$ 3. **Final answer:** $f(9) = 29$. --- 1. **Problem:** Evaluate $g(-3)$ for $g(x) = -2x^2$. 2. **Calculation:** $$g(-3) = -2(-3)^2 = -2(9) = -18$$ 3. **Final answer:** $g(-3) = -18$. --- 1. **Problem:** Evaluate $g(-6)$ for $g(x) = -2x^2$. 2. **Calculation:** $$g(-6) = -2(-6)^2 = -2(36) = -72$$ 3. **Final answer:** $g(-6) = -72$. --- 1. **Problem:** Evaluate $h(3)$ for $h(x) = -4x^2 - 2x + 5$. 2. **Calculation:** $$h(3) = -4(3)^2 - 2(3) + 5 = -4(9) - 6 + 5 = -36 - 6 + 5 = -37$$ 3. **Final answer:** $h(3) = -37$. --- 1. **Problem:** Evaluate $h(8)$ for $h(x) = -4x^2 - 2x + 5$. 2. **Calculation:** $$h(8) = -4(8)^2 - 2(8) + 5 = -4(64) - 16 + 5 = -256 - 16 + 5 = -267$$ 3. **Final answer:** $h(8) = -267$. --- 1. **Problem:** Evaluate $f(\frac{2}{3})$ for $f(x) = 3x + 2$. 2. **Calculation:** $$f\left(\frac{2}{3}\right) = 3\left(\frac{2}{3}\right) + 2 = 2 + 2 = 4$$ 3. **Final answer:** $f(\frac{2}{3}) = 4$. --- 1. **Problem:** Evaluate $g(\frac{2}{3})$ for $g(x) = -2x^2$. 2. **Calculation:** $$g\left(\frac{2}{3}\right) = -2\left(\frac{2}{3}\right)^2 = -2\left(\frac{4}{9}\right) = -\frac{8}{9}$$ 3. **Final answer:** $g(\frac{2}{3}) = -\frac{8}{9}$. --- 1. **Problem:** Evaluate $h(\frac{1}{5})$ for $h(x) = -4x^2 - 2x + 5$. 2. **Calculation:** $$h\left(\frac{1}{5}\right) = -4\left(\frac{1}{5}\right)^2 - 2\left(\frac{1}{5}\right) + 5 = -4\left(\frac{1}{25}\right) - \frac{2}{5} + 5 = -\frac{4}{25} - \frac{2}{5} + 5$$ Convert to common denominator 25: $$-\frac{4}{25} - \frac{10}{25} + \frac{125}{25} = \frac{-4 - 10 + 125}{25} = \frac{111}{25} = 4.44$$ 3. **Final answer:** $h(\frac{1}{5}) = \frac{111}{25}$ or approximately 4.44.