Functions and Graphs

1
Functions and Graphs

• Chapter 1

2
Functions and Their Properties

• 1.2

3
Function

• A function from a set D to a set R is a rule that
  assigns to every element in D a unique element in
  R. The set D of all input values is the domain
  of the function, and the set R of all output
  values is the range of the function.
• Each x leads to only one y, not two.
• Two xs can go to the same y.

4
Function Notation

• yf(x)
• Y equals f of x or the value of f at x.

5
Vertical Line TestFunction?

• A graph in the xy-plane defines y as a function
  if and only if no vertical line intersects the
  graph in more than one point.

6
Determine whether the equation defines a Function

• Algebraic Test
• Solve the equation for y. If this results in an
  expression with two possible answers, then the
  equation does not represent a function.
  Otherwise, the equation is a function.

7
Example--Determine whether the equation defines a
Function

• Y x3
• Graphical
• Graph on calculator and see if it passes the
  vertical line test.
• Algebraic
• The equation is solved for y and note that for
  any x chosen, there will be only one result for
  y. Therefore this equation is a function.

8
Example--Determine whether the equation defines a
Function

• x y2 1
• Graphically
• Solve for y, dont forget the /-.
• Check to see if it passes the VLT.
• Algebraically
• Solve for y. Notice that there will be two
  possible results for each x, so this is not a
  function.
• Try x -3
• -3 y2 1
• y2 4
• This means y is either 2 or -2. Therefore this
  is not a function.

9
Example--Determine whether the equation defines a
Function

• Seeing y raised to an even power can be a clue
  that the equation is not a function.

10
Twelve Basic Functions

• F(x) x
• F(x) x2
• F(x) x3
• F(x) ln x
• F(x) vx
• F(x) 1/x
• F(x) sin x
• F(x) cos x
• F(x) x
• F(x) int(x)
• F(X) 1/(1e-x)
• F(x) ex

11
Properties of Functions

• Domain
• Range
• Continuity
• Increasing, Decreasing, Constant
• Boundedness
• Extrema
• Symmetry
• Asymptotes
• End Behavior

12
Domain

• The domain of a function is all reals unless one
  of the following situations is encountered.
• Even Roots
• Denominator
• Logarithms
• Some Trigonometric Functions
• Piecewise Functions
• Story Problems

13
Even Roots

• No negative numbers can occur under an even root.
• Set the part under the root greater than or equal
  to zero and solve for x. How do you solve an
  inequality??
• This will be the domain if the overall function
  is an even root.

14
Denominators

• A zero in the denominator makes a function
  undefined.
• Set the denominator to zero and solve for x.
• Exclude these values from the domain. All reals
  except

15
Logarithms Have Special Domains

• yLn(x) has a domain of xgt0.
• e(y) x e is a positive number, therefore e to
  any power will be positive.
• yLoga(x) has a domain of xgt0 when agt0.
• a(y) x if a is positive, then a to any power
  will be positive. Typically a is 10.

16
Trigonometric Functions

• Some Trigonometric functions have special domains.

17
Piecewise Functions

• Have a domain given in the creation of the
  function.
• f(x) x if x0
• x2 if xgt0

18
Piecewise Functions

• Have a domain given in the creation of the
  function.
• f(x) 0 if 0xlt1
• 1 if x1
• 2 if 1ltxlt4

19
Domain

• In story problems we will use a relevant domain.
  Otherwise we will use the implied domain.( the
  domain of the given function.)

20
Example--Domain

• (2x)/(x2 4)
• This example has a denominator, so set it equal
  to zero and solve.
• x2 4 0
• X 2, -2
• Therefore the domain is all reals except 2 and -2.

21
Example--Domain

• (x)/sqrt(x 4)
• This example has a denominator, so set it equal
  to zero and solve.
• Sqrt(x 4) 0
• X 4
• It also has a square root, so set what is under
  the square root to 0.
• X-4 0, solve this
• x 4
• Therefore the domain needs to have x greater than
  or equal to 4 and not equal to 4.
• Domain is all reals greater than 4. x gt 4

22
Piecewise Functions

• Have a domain given in the creation of the
  function.
• f(x) 0 if 0xlt1
• 1 if x1
• 2 if 1ltxlt4

23
Piecewise Functions

• Have a domain given in the creation of the
  function.
• f(x) x if x0
• x2 if xgt0

24
Range

• Graph the function on the calculator.
• The range is all the y values that the calculator
  will plot.
• We will look more at this later. For now we will
  use the calculator.

25
Example--Range

• (2x)/(x2 4)

26
Example--Range

• (x)/sqrt(x 4)

27
Continuity

• A function is continuous if the entire graph can
  be traced with a pencil without lifting up the
  pencil.
• There should be no breaks or jumps.

28
Continuous at xa

• If the limit as x approaches a of f(x) equals the
  value of the function at a, then the function is
  continuous at a.
• If f(3) 5 and the limit as x approaches 3 is 5,
  then the function is continuous at 3.

29
Limit as x approaches a.

• We want to know what is happening near the point
  (a,f(a)) on the graph of the function.
• So, find the point (a-.00000001, f(a-.0000001))
• Pretend you are standing on the graph at this
  point.
• Look just to the left and see what y value the
  graph is approaching, call it b.

30
Limit as x approaches a.

• Now find the point (a.0000001,f(a.0000001)).
• Pretend you are standing on the graph at this
  point.
• Look just to the right and see what y value the
  graph is approaching, call it c.
• If b c, then the limit exists and is b.

31
Definition of a Limit
The limit of f of x as x approaches c equals L.
32
Discontinuity

• Points at which the function is not continuous.

33
Removable Discontinuity
There is a hole in the graph and a point
somewhere above or below the hole.
34
Removable Discontinuity
There is a hole in the graph.
35
Jump Discontinuity
In order to trace the entire graph, there is a
place where the pencil needs to be lifted off and
moved up a finite amount to reach another piece
of the graph to continue tracing it.
36
Infinite Discontinuity
In order to trace the entire graph, there is a
place where the pencil needs to be lifted off and
moved up an infinite amount to reach another
piece of the graph to continue tracing it.
37
Continuity Examples

• 22,24

38
Increasing

• A function is increasing on an interval if as x
  gets larger from left to right, then y gets
  larger from left to right.

39
Decreasing

• A function is decreasing on an interval if as x
  gets smaller from left to right, then y gets
  smaller from left to right.

40
Constant

• A function is constant on an interval if as x
  gets larger or smaller from left to right, then y
  stays the same.

41
Where is the function increasing, decreasing, or
constant?
42
Bounded Below

• A function is bounded below if there is some
  number b that is less than or equal to every
  number in the range of f. Any such number b is
  called a lower bound of f.
• Find a yb. If y never gets smaller than b, then
  the function is bounded below.

43
Bounded Above

• A function is bounded above if there is some
  number B that is greater than or equal to every
  number in the range of f. Any such number B is
  called an upper bound of f.
• Find a yb. If y never gets larger than b, then
  the function is bounded above.

44
Bounded

• A function is bounded if it is bounded both above
  and below.

45
Is the function bounded, bounded above, bounded
below, or unbounded?
Bounded above
46
Is the function bounded, bounded above, bounded
below, or unbounded?
Bounded below
47
Is the function bounded, bounded above, bounded
below, or unbounded?
Unbounded if ends go off to infinity, otherwise
bounded.
48
Local Maximum

• A value for the function that is greater than all
  other values in a specified open interval.

49
Absolute Maximum

• A value for the function that is greater than all
  other values in the functions entire range.

50
Local Minimum

• A value for the function that is less than all
  other values in a specified open interval.

51
Absolute Minimum

• A value for the function that is less than all
  other values in the functions entire range.

52
Extrema

• A word to cover maximums, and minimums.

53
Example--Max/Min

• WHAT is the max/min means state the y-value.
• WHERE is the max/min means state the x-value.

54
State the local max, local min, global max and
global min.
Local max
55
Symmetry

• 1. Symmetry with respect to the x-axis. If
  (x,y) is on the graph, then (x,-y) is on the
  graph. Graph folds over the x-axis to match up.
• Even Symmetry--Symmetry with respect to the
  y-axis. If (x,y) is on the graph, then (-x,y) is
  on the graph. Graph folds over the y-axis to
  match up. f(x)f(-x)

56
Symmetry

• Odd Symmetry--Symmetry with respect to the
  origin. If (x,y) is on the graph, then (-x,-y)
  is on the graph. Graph folds over the x-axis and
  then the y-axis to match up. f(-x)-f(x)

57
Symmetry Examples

• 48-54 even

58
Horizontal Asymptotes

• The line yb is a horizontal asymptote of the
  graph of a function if the function approaches a
  limit of b as x approaches infinity.
• As x gets extremely large positively, what is y?
  If it is approaching a constant, then that number
  is a horizontal asymptote for the function.

59
Horizontal Asymptotes

• The line yb is a horizontal asymptote of the
  graph of a function if the function approaches a
  limit of b as x approaches infinity.
• As x gets extremely large negatively, what is y?
  If it is approaching a constant, then that number
  is a horizontal asymptote for the function.

60
Horizontal Asymptotes

• OR

61
Vertical Asymptotes

• The line xa is a vertical asymptote of the graph
  of a function if the function approaches a limit
  of 8 or -8 as x approaches a from either
  direction.
• They can occur when the denominator of the
  function is zero. Find these values that make
  the denominator zero and then check with the
  graph.

62
Vertical Asymptotes

• OR

63
Find Vertical and Horizontal Asymptotes

• (x1)/((x1)(x-3))

3
64
Vertical and Horizontal Asymptote Examples

• 56-62

65
End Behavior

• Horizontal Asymptotes tell us end behavior for
  some functions.
• What about others?
• As x goes to infinity, y also goes to infinity,
  or negative infinity.

66
Horizontal Asymptotes, End Behavior

• Go to TBLSET, put in a large positive number for
  x, then go to the table and see what happens for
  y.
• Now try a large negative number for x, then go to
  the table and see what happens to y.
• If y approaches a constant number, then that is
  the horizontal asymptote which describes the end
  behavior.
• If y approaches positive or negative infinity,
  then that means the graph grows without bound on
  the ends.

67
Horizontal Asymptotes, End Behavior

• Examples--64,66