2.1 F15 O’Brien CA 6th ed HLR 2.1: Graphs of Basic Functions and Relations; Symmetry I. Continuity A. Informal Definition A function is continuous over an interval of its domain if its hand-driven graph over that interval can be drawn without lifting the pencil from the paper. B. Characteristics of Continuous Functions Continuous functions have no points of discontinuity (no holes, jumps, breaks, or gaps). Nor do they have a vertical asymptote. An asymptote is an imaginary line which the graph of the function gets close to but which it does not intersect. An asymptote is not part of the graph, but is used as an aid in drawing it. Example 1 The two functions below are both discontinuous. The one on the left has a point of discontinuity at x = 3. It is continuous over the intervals , 3 and 3, . And the one on the right has a vertical asymptote at x = 2. It is continuous over the intervals , 2 and 2, . II. Increasing, Decreasing, and Constant Intervals A. If a function rises from left to right on the open interval from a to b, (a, b), it is increasing on that interval. B. If a function falls from left to right on the open interval from a to b, (a, b), it is decreasing on that interval. C. If a function is horizontal from left to right on the open interval from a to b, (a, b), it is constant on that interval. Examples 2 and 3 III. Library of Functions A. Identity Function f(x) = x , , x y -2 -2 -1 -1 0 0 Decreasing Interval: None 1 1 Constant Interval: None 2 2 Domain: Range: Increasing Interval: , Continuous on , 1 2.1 F15 O’Brien CA 6th ed HLR B. f(x) = x2 Squaring Function , 0, x y -2 4 -1 1 0 0 1 1 2 4 x y -2 -8 -1 -1 0 0 1 1 2 8 Domain: 0, x y Range: 0, 0 0 1 1 4 2 9 3 16 4 x y -8 -2 -1 -1 0 0 1 1 8 2 Domain: Range: 0, Decreasing Interval: , 0 Increasing Interval: Constant Interval: None Continuous on , C. Cubing Function Domain: Range: f(x) = x3 , , Increasing Interval: , Decreasing Interval: None Constant Interval: None Continuous on , D. f(x) x Square Root Function Increasing Interval: 0, Decreasing Interval: None Constant Interval: None Continuous on 0, E. f(x) 3 x Cube Root Function Domain: Range: , , Increasing Interval: , Decreasing Interval: None Constant Interval: None Continuous on , 2 2.1 F15 O’Brien CA 6th ed HLR F. Absolute Value Function Domain: Range: f x x , 0, 0, Decreasing Interval: , 0 Increasing Interval: Constant Interval: None Continuous on , x y -2 -2 -1 1 0 0 1 1 2 1 3