1. 2.6 - 1 10 TH EDITION LIAL HORNSBY SCHNEIDER COLLEGE ALGEBRA

2. 2.6 - 2 2.6 Graphs of Basic functions Continuity The Identity, Squaring, and Cubing Functions The Square Root and Cube Root Functions The Absolute Function Piecewise-Defined Functions The Relation x = y 2

3. 2.6 - 3 Continuity (Informal Definition) A function is continuous over an interval of its domain if its hand-drawn graph over that interval can be sketched without lifting a pencil from the paper.

4. 2.6 - 4 Example 1 DETERMINING INTERVALS OF CONTINUTIY Describe the intervals of continuity for each function. Solution The function is continuous over its entire domain,(– ,  ).

5. 2.6 - 5 Example 1 DETERMINING INTERVALS OF CONTINUTIY Describe the intervals of continuity for each function. Solution The function has a point of discontinuity at x = 3. Thus, it is continuous over the intervals, (– , 3) and (3,  ). 3

6. 2.6 - 6 Domain: (– ,  ) Range: (– ,  ) IDENTITY FUNCTION  (x) = x xy – 2– 2– 2– 2 – 1– 1– 1– 1 00 11 22  (x) = x is increasing on its entire domain, (– ,  ). It is continuous on its entire domain. y x

7. 2.6 - 7 Domain: (– ,  ) Range: [0,  ) SQUARING FUNCTION  (x) = x 2 xy – 2– 24 – 1– 11 00 11 24  (x) = x 2 decreases on the interval (– ,0] and increases on the interval [0,  ). It is continuous on its entire domain, (– ,  ). y x

8. 2.6 - 8 Domain: (– ,  ) Range: (– ,  ) CUBING FUNCTION  (x) = x 3 xy – 2– 2– 8– 8 – 1– 1– 1– 1 00 11 28  (x) = x 3 increases on its entire domain, (– ,  ). It is continuous on its entire domain, (– ,  ). y x

9. 2.6 - 9 Domain: [0,  ) Range: [0,  ) SQUARE ROOT FUNCTION  (x) = xy 00 11 42 93 164  (x) = increases on its entire domain, [0,  ). It is continuous on its entire domain, [0,  ). y x

10. 2.6 - 10 Domain: (– ,  ) Range: (– ,  ) CUBE ROOT FUNCTION  (x) = xy – 8– 8– 2– 2 – 1– 1– 1– 1 00 11 82  (x) = increases on its entire domain, (– ,  ). It is continuous on its entire domain, (– ,  ). y x

11. 2.6 - 11 Domain: (– ,  ) Range: [0,  ) ABSOLUTE VALUE FUNCTION  (x) = xy – 2– 22 – 1– 11 00 11 22  (x) = decreases on the interval (– , 0] and increases on [0,  ). It is continuous on its entire domain, (– ,  ). y x

12. 2.6 - 12 Example 2 GRAPHING PIECEWISE-DEFINED FUNCTIONS Graph the function. a. b.

13. 2.6 - 13 Example 2 GRAPHING PIECEWISE-DEFINED FUNCTIONS Graph the function. a. Solution 2 4 6 – 2– 2 3 5 (2, 3) (2, 1) y x

14. 2.6 - 14 Example 2 GRAPHING PIECEWISE-DEFINED FUNCTIONS Graph the function. b. 2 4 6 – 3– 3 3 5 (1, 5) Solution y x

15. 2.6 - 15

16. Domain: (– ,  ) Range: {y  y is an integer} = {…,– 2, – 1, 0, 1, 2, 3,…} GREATEST INTEGER FUNCTION  (x) = xy – 2– 2– 2– 2 – 1.5– 2– 2 –.99– 1– 1 00.0010 33 3.993  (x) = is constant on the intervals…, [– 2, – 1), [– 1, 0), [0, 1), [1, 2), [2, 3),… It is discontinuous at all integer values in its domain (– ,  ). 1 2 3 1 2 – 2– 2 3 – 2– 2 4 – 3– 3 – 4– 4 – 3– 3 – 4– 4 4 2.6 - 16

17. 2.6 - 17 Example 3 GRAPHING A GREATEST INTEGER FUNCTION Graph Solution If x is in the interval [0, 2), then y = 1. For x in [2, 4), y = 2, and so on. Some sample ordered pairs are given here. x0½13/2234 – 1– 1– 2– 2– 3– 3 y1 11122300 –1 The ordered pairs in the table suggest a graph similar to the one in the previous slide. The domain is (– ,  ). The range is {…, – 2, – 1, 0, 1, 2,…}.

18. 2.6 - 18 Example 4 APPLYING A GREATEST INTEGER FUNCTION An express mail company charges $25 for a package weighing up to 2 lb. For each additional pound or fraction of a pound there is an additional charge of $3. Let D(x) represent the cost to send a package weighing x pounds. Graph y = D(x) for x in the interval (0, 6]

19. 2.6 - 19 Example 4 APPLYING A GREATEST INTEGER FUNCTION Solution For x in the interval (0, 2], y = 25. For x in(2, 3], y = 25 + 3 = 28. For x in (3, 4], y = 28 + 3 = 31, and so on. 0 20 Pounds 123 45 y 30 40 6 x Dollars

20. 2.6 - 20 The Relation x = y 2 Recall that a function is a relation where every domain is paired with one and only one range value. xy 00 1 11 4 22 9 33 y x Note that this is a relation, but not a function. Domain is [0,  ). Range is (– ,  ).