This document discusses techniques for illustrating inequalities between combined functions graphically and algebraically. It presents three graphical methods - comparing functions visually with different colors, analyzing the difference function by subtracting one from the other, and analyzing the quotient of the functions. The document also provides an example of identifying points of intersection between two functions, f(x) = x and g(x) = (x-2)2, graphically and algebraically to verify the solutions.

1. Inequalities of Combined Functions Example 1 TECHNIQUES FOR ILLUSTRATING INEQUALITIES Section 8.4

2. Graphing to Identify P.O.I’s Let f(x) = x and g(x) = (x-2)2

3. Graphing to Identify P.O.I From graphs we can see that P.O.I’s are (1,1) and (4,4) We can verify this algebraically:

4. Illustrating Graph Regions: Method 1 Compare the function visually Different colours represent different regions of graph Easy to visualize regions where one function is greater than the other

5. Analyze the Difference Function Subtract the functions and see where the graph of the difference is above the x-axis Illustrating Graph Regions: Method 2

7. Illustrating Graph Regions: Method 3 F(x) and g(x) can also be compared by analysing their quotient. Graph the combined function and identify the intervals for which This quotient is greater than one, which will correspond to where f(x) > g(x)

8. Conclusively, algebraic and graphical representations of inequalities can be useful for solving problems involving combined functions.