1. **Problem Statement:** Analyze the graph of a function to determine its domain, range, whether it is one-one, calculate values at specific points, explain dependent and independent variables, find rate of change, slopes, local extrema, intervals of increase/decrease, and describe piecewise functions including tax calculations. 2. **Domain and Range:** The domain is the set of all possible input values (x-values) for the function. The range is the set of all possible output values (y-values). From the graph, identify the minimum and maximum x-values for the domain and minimum and maximum y-values for the range. 3. **Function and One-One Function:** A function assigns exactly one output for each input. A one-one (injective) function means each output corresponds to exactly one input. Use the vertical line test to confirm it is a function and the horizontal line test to check if it is one-one. 4. **Calculating Values at Specific Points:** To find the function value at 70 and 20 thousand, substitute these x-values into the function or read from the graph. 5. **Dependent and Independent Variables:** The independent variable is the input (usually x), and the dependent variable is the output (usually y). The dependent variable depends on the independent variable. 6. **Rate of Change and Slopes:** The rate of change is the slope of the graph, calculated as $$\text{slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$$. Calculate slopes for both categories by choosing two points on each graph. 7. **Local Extrema:** Local maxima are points where the function changes from increasing to decreasing; local minima are where it changes from decreasing to increasing. Identify these points on the graph. 8. **Intervals of Increase and Decrease:** Determine intervals where the function is increasing (slope > 0) or decreasing (slope < 0) by analyzing the graph. 9. **Piecewise Function and Tax Calculation:** A piecewise function is defined by different expressions over different intervals. Calculate taxes at each slab by applying the corresponding tax rate to the income range. **Final Answer:** - Domain and range are identified from the graph. - The graph passes the vertical line test, so it is a function. - Use the horizontal line test to check if it is one-one. - Values at 70 and 20 thousand are calculated from the function. - Independent variable is weight; dependent variable is length. - Slopes and rate of change are calculated using two points. - Local maxima and minima are identified. - Intervals of increase and decrease are described. - Piecewise tax function is explained and taxes calculated accordingly.