1. **Task 1: Interpret the Domain and Range Graph** 1. The problem asks to identify the domain and range of the graph, and determine if it represents a function and a one-to-one function. 2. **Domain** is the set of all possible input values (x-values) for the graph. 3. **Range** is the set of all possible output values (y-values) for the graph. 4. To check if the graph represents a function, use the **vertical line test**: if any vertical line intersects the graph more than once, it is not a function. 5. To check if the function is one-to-one, use the **horizontal line test**: if any horizontal line intersects the graph more than once, it is not one-to-one. 6. Without the exact graph, generally: - Domain: all x-values covered by the graph. - Range: all y-values covered by the graph. - Function: passes vertical line test. - One-to-one: passes horizontal line test. 2. **Task 2: Avocado Export Function E(P) = P - 10000, P ≥ 10000** 1. The problem defines export $E(P) = P - 10000$ where $P \geq 10000$. 2. This is a linear function with domain $[10000, \infty)$. 3. The range is $[0, \infty)$ because when $P=10000$, $E(10000)=0$. 4. To check if $E(P)$ is a function: for each $P$ there is exactly one $E(P)$, so yes, it is a function. 5. Calculate exports for $P=70000$ and $P=20000$ (in thousands): - $E(70000) = 70000 - 10000 = 60000$ - $E(20000) = 20000 - 10000 = 10000$ 6. Independent variable: $P$ (production). 7. Dependent variable: $E(P)$ (export). 3. **Task 3: Graph of f and g with intersection at A(5,25)** 1. At intersection point A(5,25), both functions have the same value. 2. Rate of change (slope) is $\frac{\Delta y}{\Delta x}$. 3. For each function, select two points (C,D) on f and (E,F) on g, calculate slopes: - Slope of CD: $m_{CD} = \frac{y_D - y_C}{x_D - x_C}$ - Slope of EF: $m_{EF} = \frac{y_F - y_E}{x_F - x_E}$ 4. Comparing slopes gives insight into how weight changes with length for each animal. 4. **Task 4: Local Extrema Explanation** 1. Local extrema are points where the function changes direction locally. 2. Local maximum: function value is higher than nearby points. 3. Local minimum: function value is lower than nearby points. 4. Global max/min are the highest/lowest values over the entire domain. 5. Intervals of increase/decrease are where function rises/falls. 6. Identify intervals (A,B) etc. and specify increasing or decreasing. 5. **Task 5: Piecewise Tax Function for Country W** 1. Define tax function $T(x)$ where $x$ is income: $$ T(x) = \begin{cases} 0.10x & \text{if } 0 \leq x \leq 2200 \\ 0.10 \times 2200 + 0.185(x - 2200) & \text{if } 2200 < x \leq 8945 \\ 0.10 \times 2200 + 0.185 \times (8945 - 2200) + 0.30(x - 8945) & \text{if } x > 8945 \end{cases} $$ 2. Calculate tax for sample incomes: - For $x=2000$: $T(2000) = 0.10 \times 2000 = 200$ - For $x=5000$: $T(5000) = 0.10 \times 2200 + 0.185 \times (5000 - 2200) = 220 + 0.185 \times 2800 = 220 + 518 = 738$ - For $x=10000$: $T(10000) = 0.10 \times 2200 + 0.185 \times (8945 - 2200) + 0.30 \times (10000 - 8945) = 220 + 1250.325 + 316.5 = 1786.825$ **Summary:** - Task 1: Domain, range, function and one-to-one test explained. - Task 2: Linear function $E(P)$ analyzed with domain, range, function check, and values. - Task 3: Slope calculations for functions f and g at intersection and other points. - Task 4: Local extrema and intervals of increase/decrease explained. - Task 5: Piecewise tax function defined and sample tax calculations done.