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COLLEGE ALGEBRA

Learn about the graphs and continuity of identity, square, cube, square root, and cube root functions, including piecewise-defined functions. Understand intervals of continuity and application of greatest integer function in real-life scenarios.

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COLLEGE ALGEBRA

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  1. COLLEGE ALGEBRA LIAL HORNSBY SCHNEIDER

  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 = y2

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

  5. DETERMINING INTERVALS OF CONTINUTIY Example 1 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. Domain: (–, ) Range: (–, ) IDENTITY FUNCTION  (x) = x y x (x) = xis increasing on its entire domain, (–, ). It is continuous on its entire domain.

  7. Domain: (–, ) Range: [0, ) SQUARING FUNCTION  (x) = x2 y x (x) = x2 decreases on the interval (–,0] and increases on the interval [0, ). It is continuous on its entire domain,(–, ).

  8. Domain: (–, ) Range: (–, ) CUBING FUNCTION  (x) = x3 y x (x) = x3 increases on its entire domain, (–,) . It is continuous on its entire domain,(–, ).

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

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

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

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

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

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

  15. GREATEST INTEGER FUNCTION  (x) = Domain: (–, ) Range: {y  y is an integer} = {…,–2, –1, 0, 1, 2, 3,…} 4 3 2 1 –4 –3 –2 1 4 3 2 –2 –3 –4 (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 (–, ). 2.6 - 16

  16. GRAPHING A GREATEST INTEGER FUNCTION Example 3 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. 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,…}.

  17. APPLYING A GREATEST INTEGER FUNCTION Example 4 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]

  18. APPLYING A GREATEST INTEGER FUNCTION Example 4 y 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. 40 30 Dollars 20 x 0 1 2 3 4 5 6 Pounds

  19. The Relation x = y2 Recall that a function is a relation where every domain is paired with one and only one range value. y x Note that this is a relation, but not a function. Domain is [0, ). Range is (–, ).

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