1. **Exercise (1): Use vertical line test to verify if each graph is a function.** - The vertical line test states that if any vertical line intersects the graph more than once, the graph is not a function. 1) The first graph is a piecewise linear shape increasing then decreasing. Any vertical line will intersect it only once. **Conclusion:** It is a function. 2) The second graph has three rays crossing at the origin forming a star shape. Vertical lines near the origin intersect the graph more than once. **Conclusion:** It is not a function. 3) The third graph is a half-ellipse or semicircle opening right. Vertical lines inside the semicircle intersect twice. **Conclusion:** It is not a function. 2. **Exercise (2): Determine the Domain.** 1) Table with x-values 1,4,5 and corresponding y-values -2,8,4. **Domain:** The set of x-values given: $\{1,4,5\}$. 2) Increasing curve from origin upwards. **Domain:** Since it starts at origin and increases, domain is $[0, \infty)$. 3) $g(x) = \frac{4}{2x-8}$ Domain excludes values making denominator zero: Set denominator to zero: $2x-8=0 \Rightarrow x=4$ **Domain:** $\{x \in \mathbb{R} | x \neq 4\}$. 4) $g(x) = 7\sqrt{6x - 42}$ Inside the square root must be non-negative: $6x - 42 \geq 0 \Rightarrow 6x \geq 42 \Rightarrow x \geq 7$ **Domain:** $[7, \infty)$. 3. **Exercise (3): Evaluate given functions.** 1) $p(x) = 3x^2 - 4x + 2$, find $p(-1)$: Calculate: $p(-1) = 3(-1)^2 - 4(-1) + 2 = 3(1) + 4 + 2 = 9$ 2) $g(x) = \frac{4x^3 - 2}{2x^{7/4}}$, find $g(0)$: Calculate denominator at $x=0$: $2 \cdot 0^{7/4} = 0$, division by zero undefined. **Conclusion:** $g(0)$ is undefined. 3) $h(x) = 4x - 8 + x^3$, find $h(3)$: Calculate: $h(3) = 4(3) - 8 + 3^3 = 12 - 8 + 27 = 31$ 4. **Exercise (4): Reflect points across axes and origin.** Reflection rules: - Across x-axis: $(x,y) \to (x, -y)$ - Across y-axis: $(x,y) \to (-x, y)$ - Across origin: $(x,y) \to (-x, -y)$ Points: 1) $A(3,6)$ - x-axis: $(3,-6)$ - y-axis: $(-3,6)$ - origin: $(-3,-6)$ 2) $B(-2,8)$ - x-axis: $(-2,-8)$ - y-axis: $(2,8)$ - origin: $(2,-8)$ 3) $C(6,-19)$ - x-axis: $(6,19)$ - y-axis: $(-6,-19)$ - origin: $(-6,19)$ 4) $D(-5,-3)$ - x-axis: $(-5,3)$ - y-axis: $(5,-3)$ - origin: $(5,3)$ 5. **Exercise (5): Graph and give domain and range.** 1) $R(x) = (x+2)^2$ - Domain: All real numbers $(-\infty, \infty)$ - Range: Since square is always non-negative, minimum at $x=-2$ is 0, so range is $[0, \infty)$ 2) $g(x) = (x-1)^3$ - Domain: All real numbers $(-\infty, \infty)$ - Range: All real numbers $(-\infty, \infty)$ because cubic function covers all y-values 3) $h(x) = \sqrt{x-4}$ - Domain: $x-4 \geq 0 \Rightarrow x \geq 4$, so $[4, \infty)$ - Range: Square root outputs non-negative values, so $[0, \infty)$