Inequalities of Combined Functions Example 1

TECHNIQUES FOR ILLUSTRATING INEQUALITIES

Section 8.4

Graphing to Identify P.O.I’s

• Let f(x) = x and g(x) = (x-2)2

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:

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

• 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

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)

Conclusively, algebraic and graphical representations of inequalities can be useful for solving problems involving

combined functions.