Sect 2.6 Graphs of Basic Functions

Transcription

1 Sect. Graphs of Basic Functions Objective : Understanding Continuity. Continuity is an extremely important idea in mathematics. When we say that a function is continuous, it means that its graph has no holes or gaps. In other words, a function f is continuous at a point c if you can draw the graph through the point (c, f(c)) without lifting your pencil. Informal Definition of Continuity A function is continuous over an interval of its domain if for every point (c, f(c)) in that interval, you can draw the graph of f through that point without lifting your pencil. Ex. f x Here, the function is continuous at every point since there are no holes or gaps in the graph. So, we can say f is continuous on its domain. Ex. a) b) c) f f f c x c x c x In each of the examples, f is not continuous at the point (c, f(c)). We can say that f is discontinuous at x = c.

2 Determine the intervals where the function is continuous: Ex. Ex Solution: Solution: We can draw the graph We have to lift our pencil when we through every point without draw the graph at x =, so the lifting our pencils so the function is discontinuous at x =. function is continuous on We can say the function is (, ). continuous on (, ) U (, ) Objective : Graphs of Basic Functions. Ex. f(x) = x Identity Function x f(x) = x Points (, ) (, ) (, ) (, ) (, ) (, ) (, ) Domain: (, ) Range: (, ) f is continuous on (, ). f is increasing on (, ). f is decreasing nowhere.

3 Ex. f(x) = x Squaring Function x f(x) = x Points () = (, ) () = (, ) () = (, ) () = 9 (, 9) ( ) = (, ) ( ) = (, ) ( ) = 9 (, 9) f(x) = x Domain: (, ) Range: [, ) f is continuous on (, ). f is increasing on [, ). f is decreasing on (, ]. The graph is called a parabola. The point (, ) is called the vertex. Ex. f(x) = x Cubing Function x f(x) = x Points () = (, ) () = (, ) () = 8 (, 8) () = (, ) ( ) = (, ) ( ) = 8 (, 8) ( ) = (, ) Domain: (, ) Range: (, ) f is continuous on (, ). f is increasing on (, ) f is decreasing nowhere. f(x) = x The point (, ) where f changes from opening downward to opening upward is called an inflection point.

4 Ex. 8 f(x) = x Square Root Function x f(x) = x Points = (, ) = (, ) = (, ) 9 9 = (9, ) 9 is not a real # is not a real # 9 is not a real # No point No point No point 9 8 f(x) = x Domain: [, ) Range: [, ) f is continuous on [, ). f is increasing on [, ). f is decreasing nowhere. Ex. 9 f(x) = x x f(x) = x Cube Root Function Points = (, ) = (, ) = (8, ) = (, ) = (, ) = ( 8, ) = (, ) Domain: (, ) Range: (, ) f is continuous on (, ). f is increasing on (, ) f(x) = f is decreasing nowhere. x

5 Ex. f(x) = x Absolute Function x f(x) = x Points = (, ) = (, ) = (, ) = (, ) = (, ) = (, ) = (, ) f(x) = x Domain: (, ) Range: [, ) f is continuous on (, ). f is increasing on [, ). f is decreasing on (, ]. Ex. f(x) = x Reciprocal Function x.. f(x) = x Points is undef. None. = (., ) = (, ) =. (,.). = (., ) = (, ) =. (,.) f(x) = x Domain: (, ) U (, ) Range: (, ) U (, ) f is continuous on (, ) U (, ). f is increasing nowhere. f is decreasing on (, ) U (, ).

6 Objective : Graphing Piecewise Defined Functions To graph a piecewise defined function, we graph each piece and then take the parts of each of the graph for which the function is defined and splice them together. Graph the following: Ex. { x if x f(x) = x + if < x x if x > Solution: The first piece (x ) is the graph of y = x reflected across the x axis and shifted up vertically by units. The second piece ( < x ) is a line with y-intercept of (, ) and slope of. The third piece is the graph of y = x shifted down vertically by units. Then, we select the appropriate pieces to construct our graph: x < x x > Splicing our pieces together, we get:

7 8 A type of piecewise defined function that occurs in many applications is the "step function." One type of "step functions" is the greatest integer function is defined as [[x]] = greatest integer less than or equal to x. In other words, if x is an integer, then [[x]] = x. If x is not an integer, then [[x]] will equal the integer immediately to the left of x. Thus, [[]] =, [[.8]] =, [[.999]] =, [[.]] =, and [[.]] =. Graph the following Ex. f(x) = [[x]] Solution: When x is between two consecutive integers, its value is constant, giving us a graph that is a series of horizontal line segments. x f(x) = [[x]] x < x < x < x < x < Domain: (, ) Range: {,,,,,, } f is discontinuous at {,,,,,, } f is constant on the intervals [, ), [, ), [, ), [, ), A classic example of a step function is postage rates. For a letter below one ounce, it costs 9 to mail. For each additional ounce, it costs per ounce, so from ounce to just below ounces, it will cost to mail. So, our function is: Cost in Cents Number of Ounces - - -

8 The last graph we are going to examine is the graph of x = y. Notice that this is a relation and not a function since y = ± x. Ex. x = y x y = ± x Points (, ) ± = ± ± = ± 9 ± 9 = ± (, ) & (, ) (, ) & (, ) (9, ) & (9, ) Domain: [, ) Range: (, ) - The graph is a parabola that is turned on - its side opening to the right. In order to generate its graph on a graphing calculator, you will need to graph two separate pieces at the same time. If you let Y = x and Y = x, and graph both of them, you will get the parabola above.