1.3 twelve basic functions basic function:y=x, linear …€¦ · local extrema every non integer...
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1.3 12 functions complete.notebook August 31, 2017
1.3 Twelve Basic Functions
In your notes, copy the following table:
Basic Function: y=x, linear functionGraph:
Domain (∞,∞)Range(∞,∞)Continuity yesIncreasing/decreasing behavior Inc (∞,∞)Symmetry yes, about the origin, odd functionLocal Extrema NoneHorizontal asymptotes NoneVertical asymptotes None
1.3 12 functions complete.notebook August 31, 2017
Basic Function: y=x2Graph:
Domain (∞,∞)Range [0,∞)Continuity yesIncreasing/decreasing behavior
dec on (∞,0] inc on [0,∞)Symmetry yes about the yaxis even functionLocal Extrema yes min (0,0)Horizontal asymptotes noneVertical asymptotes none
End behavior
1.3 12 functions complete.notebook August 31, 2017
Basic Function: y=x3 cubic Graph:
Domain (∞,∞)Range (∞,∞)Continuity yes
Increasing/decreasing behavior inc (∞,∞)Symmetry yes about the origin odd functionLocal Extrema noneHorizontal asymptotes noneVertical asymptotes none
End behavior
1.3 12 functions complete.notebook August 31, 2017
Basic Function: y=|x|Graph:
Domain (∞,∞)Range [0,∞)Continuity yesIncreasing/decreasing behavior
dec (∞,0] inc [0,∞)Symmetry yes about the yaxis even functionLocal Extrema min @ (0,0) Horizontal asymptotes noneVertical asymptotes noneEnd behavior
1.3 12 functions complete.notebook August 31, 2017
Basic Function: y=1/x reciprocalGraph:
Domain (∞,0)U(0,∞)Range (∞,0)U(0,∞)Continuity no, infinite discontinuity @x=0Increasing/decreasing behavior
dec on (∞,0) dec on (0,∞)Symmetry yes about the origin odd functionLocal Extrema noneHorizontal asymptotes yes y=0Vertical asymptotes yes x=0End behavior
1.3 12 functions complete.notebook August 31, 2017
Basic Function: y=√x square rootGraph:
Domain [0,∞)Range [0,∞)Continuity yesIncreasing/decreasing behavior inc [0,∞)Symmetry noneBoundedness below by y=0Local Extrema (0,0) absolute minimumHorizontal asymptotes noneVertical asymptotes none
End behavior
1.3 12 functions complete.notebook August 31, 2017
Basic Function: y=ex exponentialGraph:
Domain (∞,∞)Range (0,∞)Continuity yes
Increasing/decreasing behavior inc (∞,∞)Symmetry noneBoundedness below by y=0Local Extrema noneHorizontal asymptotes yes y=0Vertical asymptotes none
End behavior
1.3 12 functions complete.notebook August 31, 2017
Basic Function: y=ln(x)Graph:
Domain (0,∞)Range (∞,∞)Continuity yes
Increasing/decreasing behavior inc (0,∞)Symmetry noLocal Extrema noneHorizontal asymptotes noneVertical asymptotes yes, x=0
End behavior
1.3 12 functions complete.notebook August 31, 2017
Basic Function: y=sin(x)Graph:
Domain (∞,∞)Range [1,1]Continuity yesIncreasing/decreasing behavior lots more laterSymmetry yes about the origin odd functionBoundedness yes above by y=1, below by y= 1Local Extrema yes several maxes and mins (more later)Horizontal asymptotes noneVertical asymptotes noneEnd behavior None or DNE
1.3 12 functions complete.notebook August 31, 2017
Basic Function: y=cos(x)Graph:
Domain (∞,∞)Range [1,1]Continuity yesIncreasing/decreasing behavior more later Symmetry yes about the yaxis even functionBoundedness yes, above by y=1, below by y= 1Local Extrema yes several maxes and mins (more later) Horizontal asymptotes noneVertical asymptotes noneEnd behavior none
1.3 12 functions complete.notebook August 31, 2017
Basic Function: y=[x] or Int[x]Graph:
Domain (∞,∞)Range all integersContinuity no discontinuity at each integer value of xIncreasing/decreasing behavior constant on intervals of the form [k,k+1)Symmetry noneLocal Extrema every non integer value is both a local max and a local minHorizontal asymptotes noneVertical asymptotes noneEnd behavior
1.3 12 functions complete.notebook August 31, 2017
Basic Function: y=1/(1+ex)Graph:
Domain (∞,∞)Range (0,1)Continuity yesIncreasing/decreasing behavior inc (∞,∞)Symmetry noBoundedness yes, below by y=0, above by y=1Local Extrema noneHorizontal asymptotes yes, y=1 and y=0Vertical asymptotes none
End behavior
1.3 12 functions complete.notebook August 31, 2017
PIECEWISE FUNCTIONS
Ex.1Sketch a graph of the following piecewise functions.
if if
Which of the twelve basic functions makes up the piecewise
definition above?
Ex.2
Sketch a graph of the following piecewise function:
if
TRY Sketch a graph of the following piecewise function:
TRY Sketch a graph of the following piecewise function: