RELATIONS AND FUNCTIONS by gph1reD0

VIEWS: 5 PAGES: 29

									    Functions
ƒ(x) Function Notations
  A relation is a pairing between two sets. A
function is a relation in which each x-value has
                 only one y-value

Functions can be represented in many ways including
            tables, graphs and equations.

  Take for example the equation y = 2x - 3. This
    equation has an important characteristic. For
   each value of x, you find exactly one value of y.
           Relation:
 A relation is simply a set of
        ordered pairs.
 A relation can be any set of
  ordered pairs. No special
       rules need apply.
The following is an example of
          a relation:

  {(1,2)(2,4)(3,5)(2,6)(1,-3)}

The graph at the right shows
    that a vertical line may
intersect more than one point
         in a relation.
Function: A function is a set
  of ordered pairs in which
each x-value has only ONE y-
   value associated with it.
     The relation we just
           discussed
 {(1,2),(2,4,)(3,5)(2,6)(1,-3)}

  is NOT a function because
the x-value 2 is paired with a
y-value of 4 and 6. Similarly,
the x-value of 1 is paired with
    the y-value of 2 and -3
 The previous relation can
  be altered to become a
 function by removing the
ordered pairs where the x-
    value is used twice.

 Function: {(1,2)(2,4)(3,5)}

    The graph at the left
  shows that a vertical line
 intersects only ONE point
in a function. This is called
  the vertical line test for
         functions.
Function – an input-output relationship that has
exactly one output for each input.
Domain – the set of all input (x)values of a
function.
Range – The set of all output (y)values in a
function.
Function notation – the notation used to describe
a function.
      Example f(x) is read “f of x.”
                f(1) is read “f of 1.”

Linear function – a function whose graph is a
straight line.
 To determine if a relationship is a function,
verify that each input has exactly one output.
 Using tables is one way to verify functions.
Look at the function below. Can you determine
        if it is a relation or a function?
 You can identify functions using tables or
graphs. The graph below has more than one
 output for each input. Is this a function?
  A relation can be represented by a set of
 order pairs (x, y) . The first number, x, is a
member of the domain and the second number,
         y, is a member of the range.



Determine whether the order pairs make a function.

            {(-1, 7), (0, 3), (1, 5), (0, -3)}

            {(0, 2), (2, 4), (4, 8), (8, 10)}
       Vertical-Line Test for a function
   If no vertical line in the coordinate plane
intersects a graph in more than one point, then
       the graph represents a function.
       (You can use a pencil held vertically to test)
              Evaluating Functions


For the function y = 2x - 1, find f(0), f(2), and f(-1).
                     y = 2x – 1
                  f(x) = 2x -1 Write in function notation.
                  f(0) = 2(0) – 1 = -1
                  f(2) = 2(2) – 1 = 3
                 f(-1) = 2(-1) – 1 = -3
   Find f(1), f(2), f(3), and f(4).
Read the graph to find y for each x.
               f(x) = y
               f(1) = 8
              f(2) = 10
              f(3) = 12
              f(4) = 14
Linear Functions
   Straight Lines
 Writing equations of functions
 Use the equation f(x) = mx + b.
    Find b (y-intercept) = -4
Locate a point on the line, such as
              (2, 0).
 Substitute the values into your
            equation.
          f(x) = mx + b
           0 = m(2) – 4
            0 = 2m – 4
        0 + 4 = 2m -4 + 4
              4 = 2m
              2 2
               m=2
          f(x) = 2x - 4
             Writing an equation using a table

           The y-intercept can be identified from
                       the table, (0, 1)
X    Y     Pick a point, (1, 3) and substitute your
          point and y-intercept into your equation.
-2   -3
                         f(x) = mx + b
-1   -1                   3 = m(1) + 1
0    1
                           3=m+1
                       3–1=m+1–1
1    3                        m=2
2    5
                        f(x) = 2x + 1
                 Physical Science
The relationship between the two temperatures in
               the table are linear.

   Write a rule for Fahrenheit temperature as a
          function of Celsius temperature.
f(x) = mx + b, where x is Celsius and y is Fahrenheit

     Temperature (°C)   Temperature (°F)
           -10                14
            0                 32
           25                 77
           50                122
          100                212
            Practice with Functions


1)   Which of the relations below is a function?
     a) {(2, 3), (3, 4), (5, 1), (2, 4)}
     b) {(2, 3), (3, 4), (6, 2), (7, 3)}
     c) {(2, 3), ( 3, 4), (6, 2), (3, 3)}
2) Given the relation A = {(5, 2), (7, 4), (9, 10), (x, 5)}.
     Which of the following values for x will make
     relation A a function?

      a) 7
      b) 9
      c) 4
3)   The following relation is a function.
     {(10, 12), (5, 3), (15, 10), (5, 6), (1, 0)}


• True

• False
4) Which of the relations below is a function?



a)   {(1, 1,), (2, 1), (3, 1), (4, 1), (5, 1)}
b)   {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5)}
c)   {(0, 2), (0, 3), (0, 4), (0, 5), (0, 6)}
5)   The graph of a
     relation is shown at
     the right. Is this
     relation a function?

     a) yes
     b) no
     c) Cannot be
     determine from a
     graph
6) Is the relation depicted in the table below
a function?

a) yes
b) no
c) cannot be determined from a table


   X    0    1   3    5    3   9
   Y    8    9   10   6   10   7
7)   The graph of a relation is shown below.
     Is the relation a function?
     a) yes
     b) no
     c) cannot be determined from a graph
8) Is the relation in the table below a
      function?
      a) yes
      b) no



  x      -2      -1     0       1         2   3
  y       5      5      5       5         5   5
9)   The graph of a relation is shown below.
     Is the relation a function?
     a) yes
     b) no
     c) cannot be determined from a graph
10) The graph of a relation is shown below. Is
      the relation a function?
      a) yes
      b) no
      c) cannot be determined from a graph
11) Given f(x) = 3x + 7, find f(5).


              a)   15
              b)   22
              c)   42
12)   Which graph represents a function?

								
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