1. MAT 150 Unit 2-3: Piecewise, Absolute and Power Functions

2. Objectives  Evaluate and graph piecewise-defined functions  Graph the absolute function  Solve absolute value equations  Evaluate and graph power functions

3. Piecewise-Defined Functions Sometimes a function is defined in “pieces” for different intervals of the domain.

4. Evaluating and Graphing a Piecewise- Defined Function a. Evaluate f(-2), f(-1) and f(.5) b. Graph the given piecewise- defined function. c. What is the domain of the function?

5. Example The postage paid for a first-class letter “jumps” by 20 cents for each ounce after the first ounce but does not increase until the weight increases by 1 ounce. The table gives the postage for letters up to 3.5 ounces. Write a piecewise-defined function that models the price of postage and graph the function. Solution

6. Example The postage paid for a first-class letter “jumps” by 20 cents for each ounce after the first ounce but does not increase until the weight increases by 1 ounce. The table gives the postage for letters up to 3.5 ounces. Write a piecewise-defined function that models the price of postage and graph the function. Solution

7. Example The cost of weekend service calls by Airtech Services is shown by the graph. Write a piecewise-defined function that represents the cost for service during the first five hours, as defined by the graph. Solution

8. Creating a Piecewise-Defined Function The 2010 U.S. federal income tax owed by a marred couple filing jointly can be found from the following table. If Taxable Income Is Between Tax Due isOf the Amount over $0-16,75010%$0 $16,750-68,000$1675+15%$16,750 $68,000-137,000$9362.50+25%$68,000 $137,000-209,250$36,687.50+28%$137,300 $209,250-373,650$46,833.50+33%$209,250 $373,650>$101,085.50+35%$373,650 a. Write the piece-wise defined function T with input x that models the federal tax dollars owed as a p function of x, the taxable income dollars earned. b. Use the function to find T(42,000). c. Find the tax owed on a taxable income of $65,000. d. A friend tells Jack Waddell not to earn anything over $68,000 because it would raise his tax rate to %25 on all of his taxable income. Test this statement by finding the tax on $68,000 and $68,000 + $1. What do you conclude?

10. Absolute Value Equation If |x| = a and a > 0, then x = a or x = -a. There is no solution to |x| = a if a < 0; |x| = 0 has solution x = 0.

11. Solving Absolute Values

12. Power Functions A power function is a function of the form y = ax b, where a and b are real numbers, b  0.

13. Example At 17 feet, the wingspan of a prehistoric bony-toothed bird may exceed that of any other bird that ever existed. The animal weighed about 64 pounds and soared the Chilean skies 5–10 million years ago. The wingspan of a bird can be estimated using the model L = 2.43W 0.3326 where L is the wingspan in feet and W is the weight in pounds. a. Graph this function in the context of the application. b. Use the function to compute the wingspan of a mute swan that weighs 40 pounds. c. Use the function to approximate the weight of an albatross whose wingspan is 11 feet 4 inches.

14. Example (cont) Solution a. Graph this function in the context of the application. Graph the function in the first quadrant only. Since wingspan and weight must be nonnegative.

15. Example (cont) Solution b. Use the function to compute the wingspan of a mute swan that weighs 40 pounds.

16. Example (cont) Solution c. Use the function to approximate the weight of an albatross whose wingspan is 11 feet 4 inches.