This document contains solutions to homework questions about inequalities of combined functions. Question 1 discusses intervals where f(x) and g(x) are greater than each other based on a graph. Question 2 has the student graph f(x) and g(x) and state the same intervals using the graph. Question 3 solves Question 2 algebraically. Question 4 provides a word problem about a birdhouse seller's weekly costs and revenues, writing the relevant equations and expressing the profit condition. It then calculates the number of birdhouses that should be built for a profit. Question 5 changes an assumption from a previous question about variable costs, and asks how this affects the minimum and maximum number of houses that can be built.

1. INEQUALITIES OF COMBINED FUNCTIONSHOMEWORK

2. QUESTION # 1Observe the following graph. State the intervals where:F(x)>g(x) and g(x)>f(x)

3. SOLUTION # 1f(x)>g(x) when x<0, 0<x<1 and g(x)>f(x) when x>1

4. QUESTION # 2Using Graphing Technology (Ti-83, GraphCalc, Online Graphing calculator.) Graph the function f(x) and g(x) on the same set of axes. State where f(x)>g(x) and where g(x)>f(x) by looking at the graph.𝑓𝑥=𝑥−2, 𝑔𝑥=(𝑥−2)4

5. SOLUTION # 2F(x)>g(x) on the interval (2,3) and g(x)>f(x) (-∞,2) and (3,∞)

6. QUESTION # 3Solve question # 2 using a different method learned in class. Only state where f(x)>g(x)

7. SOLUTION # 3𝑓𝑥>𝑔(𝑥)𝑓𝑥−𝑔𝑥>0𝑥−2𝑥4−8𝑥3+24𝑥62−32𝑥+16>0𝑥−2−𝑥4+8𝑥3−24𝑥2+32𝑥−16>0−𝑥4+8𝑥3−24𝑥2+33𝑥−18>0 Synthetic division(𝑥−3)(−𝑥3+5𝑥−9𝑥+6) (𝑥−2)(𝑥−3)(−𝑥2+3𝑥−3) ∴ f(x)>g(x) on the interval (2,3)Cant be factored𝑥=2, 𝑥=3

8. QUESTION # 4Claire builds and sells birdhouses. Claire makes n birdhouses in a given week and sells them for 45-n dollars per birdhouse. Her weekly costs include a fixed cost of $280 plus $8 per birdhouse made. Assume that Claire sells all of the birdhouses that she makes. A) Write an equation to represent her total weekly cost.B) Write an equation to represent her total weekly revenue. C) Write an inequality to express the condition for which Claire will make a profit.D) How many birdhouses should Claire build each week in order to make a profit.

9. SOLUTION # 4C(n)=(280+8n)R(n)=(45-n)nR(n)>C(n)45−𝑛𝑛>280+845𝑛−𝑛2−8𝑛−280>0−𝑛2+37𝑛−280>0𝑛=37±372−4(−1)(−280)2(1)𝑛=37±2492𝑛=26, 𝑥=11∴11≤n≤26

10. QUESTION # 5Suppose that the developer from Question # 5, found a way to reduce their variable cost to $58 000 per houseWhat would the new cost equation be?How would this effect the minimum and maximum number of houses the developer could build.

11. SOLUTION # 6C(n)=0.058x+8It increase the maximum and decreases the minimum