Section 2.4 Library of Functions; Piecewise-Defined Functions

Transcription

1 Section. Library of Functions; Piecewise-Defined Functions Objective #: Building the Library of Basic Functions. Graph the following: Ex. f(x) = b; constant function Since there is no variable x in the equation, we can make x whatever we want and f(x) will still be b: x y b b b Plotting the points and drawing a line, we get a horizontal line. Range: {b} The function is even x-intercepts if b : None The function has absolute x-intercepts if b = : {(x, ) for any x} maxima and minima of b y-intercept: (, b) for all values of x. f is constant on (, ) Ex. f(x) = x; identity function x y Range: (, ) The function is odd The function has no absolute extrema x-intercept: (, ) y-intercept: (, ) f is increasing on (, ) b

2 Ex. f(x) = x, square function x f(x) = x Points () = (, ) () = (, ) () = (, ) () = 9 (, 9) ( ) = (, ) ( ) = (, ) ( ) = 9 (, 9) The function is even. Range: [, ) It has an absolute minimum of at. x-intercept: (, ) The function is decreasing on (, ) y-intercept: (, ) The function is increasing on (, ). Ex. f(x) = x, cube function x f(x) = x Points () = (, ) () = (, ) () = 8 (, 8) () = 7 (, 7) ( ) = (, ) ( ) = 8 (, 8) ( ) = 7 (, 7) f(x) = x The function is odd. Range: (, ) It has no absolute extrema. x-intercept: (, ) The function is increasing on (, ) y-intercept: (, )

3 Ex. f(x) = x, absolute value function x f(x) = x Points = (, ) = (, ) = (, ) = (, ) = (, ) = (, ) = (, ) The function is even. Range: [, ) It has an absolute minimum of at. x-intercept: (, ) The function is decreasing on (, ) y-intercept: (, ) The function is increasing on (, ). Ex. 6 f(x) = x, square root function x f(x) = x Points = (, ) = (, ) = (, ) 9 9 = (9, ) Not a real # No point Not a real # No point 9 Not a real # No point Domain: [, ) The function is neither even nor odd. Range: [, ) It has an absolute minimum of at. x-intercept: (, ) The function is increasing on (, ) y-intercept: (, )

4 6 Ex. 7 f(x) = x, cube root function x f(x) = x Points = (, ) = (, ) = (8, ) = (7, ) = (, ) = ( 8, ) = ( 7, ) Domain: (, ) The function is odd. Range: (, ) It has no an absolute extrema. x-intercept: (, ) The function is increasing on (, ) y-intercept: (, ) Ex. 8 f(x) =, reciprocal function x x f(x) = /x Points undefined No Point. /. = (., ) / = (, ) / =. (,.). /(.) = (., ) /( ) = (, ) /( ) =. (,.) Domain: (, ) U (, ) Range: (, ) U (, ) x-intercept: None y-intercept: None The function is odd. It has no an absolute extrema. The function is decreasing on (, ) U (, ).

5 The last basic function we need to examine is the greatest integer function or sometimes called a step function. The greatest integer function is defined as: int(x) = greatest integer less than or equal to x. Thus int() =, int(.8) =, int(.999) =, int(.7) =, & int(.) =. You can also see int(x) written as [x] or [[x]]. Graph the following Ex. 9 f(x) = int(x); greatest integer function 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) = int(x) x < x < x < x < - x < - Domain: (, ) The function is neither even nor odd. Range: { y y is an integer} It has no an absolute extrema. x-intercept: (, ) It is constant on every interval of the y-intercept: (, ) form (k, k + ) where k is an integer. A function is continuous at a point of a graph if there is no break or holes, and the graph can be drawn through that point without lifting a pencil. Thus, with the greatest integer function, it is not continuous at x = any integer since at every integer, there is a break. We say that the function is discontinuous at { x x is an integer}. 7 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.

6 Graph the following: Ex. { x if x f(x) = x + if < x x if x > 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: Range: (, ) x-intercepts: (, ) and (., ) y-intercepts: (, ) The function has no absolute extrema. It is neither even nor odd. It is increasing on (, ) U (, ). It is decreasing on (, ). It is discontinuous at x =. Ex. Ozzie s Charter-A-Bus Service has adopted the following pricing policy for groups that wish to charter its buses. For groups that contain no more than people, they will pay a flat fee of $. In groups with between and 88 people, everyone will pay $6

7 minus cents for each person in excess of. In groups with 88 or more people, everyone pays $. a) Express the bus company s revenue as a function of the size of the group. b) What is the charge for people? c) What is the charge for 78 people? a) Let x = the number of people If x, then R(x) = $. If x 88, then R(x) = x. For < x < 88, lets create a table of values: x R(x) 6.() = (6.())() 6() = (6.())() 6.() = (6.())() 6() = (6.())() x (6.(x ))(x) Thus, R(x) = (6.(x ))(x) = (6.x + )x =.x + 89x. We can then write R(x) as a piecewise defined function: { $, x R(x) =.x + 89x, < x < 88 x, x 88 b) Since x =, then R(x) = $. So R() = $. The charge is $. c) Since < x = 78 <, R(x) =.x + 89x =.(78) + 89(78) = + 69 = 9 The charge is $9. The graph of the function is given on the right. 9