TH EDITION LIAL HORNSBY SCHNEIDER COLLEGE ALGEBRA

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

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.

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

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

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

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

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

Domain: [0,  ) Range: [0,  ) SQUARE ROOT FUNCTION  (x) = xy  (x) = increases on its entire domain, [0,  ). It is continuous on its entire domain, [0,  ). y x

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

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

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

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

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

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–  (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 (– ,  ) – 2– 2 3 – 2– 2 4 – 3– 3 – 4– 4 – 3– 3 – 4–

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 y –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,…}.

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]

Example 4 APPLYING A GREATEST INTEGER FUNCTION Solution For x in the interval (0, 2], y = 25. For x in(2, 3], y = = 28. For x in (3, 4], y = = 31, and so on Pounds y x Dollars

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 (– ,  ).