Introduction to Functions

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CHAPTER 8

• Introduction to Functions

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SECTION 8-1

• Equations in Two Variables

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DEFINITIONS

• Open sentences in two variables equations and
  inequalities containing two variables

4
EXAMPLES

• 9x 2y 15
• y x2 4
• 2x y 6

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DEFINITIONS

• Solution is a pair of numbers (x, y) called an
  ordered pair.

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Example

• State whether each ordered pair is a solution of
  4x 3y 10
• (4, -2) (-2, 6)

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DEFINITIONS

• Solution set is the set of all solutions
  satisfying the sentence.

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EXAMPLE

• Solve the equation
• 9x 2y 15
• if the domain of x is
• -1,0,1,2

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SOLUTION
x (15-9x)/2 y Solution
-1 15-9(-1)/2 12 (-1,12)
0 15-9(0)/2 15/2 (0,15/2)
1 15-9(1)/2 3 (1,3)
2 15-9(2)/2 -3/2 (2,-3/2)
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SOLUTION

• ?the solution set is
• (-1,12), (0, 15/2), (1,3),
• (2,-3/2)

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EXAMPLE

• Roberto has 22. He buys some notebooks costing
  2 each and some binders costing 5 each. If
  Roberto spends all 22 how many of each does he
  buy?

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SOLUTION

• n number of notebooks
• b number of binders
• (n and b must be whole numbers)
• 2n 5b 22
• n (22-5b)/2

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SOLUTION
b (22-5b)/2 n Solution
0 22-5(0)/2 11 (0,11)
2 22-5(2)/2 6 (2,6)
4 22-5(4)/2 1 (4,1)
6 12-5(6)/2 -4 Impossible
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SOLUTION

• ?the solution set is
• (0,11), (2, 6), (4,1)

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SECTION 8-2

• Points, Lines and Their Graphs

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• COORDINATE PLANE consists of two perpendicular
  number lines, dividing the plane into four
  regions called quadrants

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• X-axis (abscissa)- the horizontal number line
• Y-axis (ordinate) - the vertical number line
• ORIGIN - the point where the x-axis and y-axis
  cross

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DEFINITION

• ORDERED PAIR - a unique assignment of real
  numbers to a point in the coordinate plane
  consisting of one x-coordinate and one
  y-coordinate

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DEFINITION

• GRAPH is the set of all points in the
  coordinate plane whose coordinates satisfy the
  open sentence.

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• LINEAR EQUATION
• is an equation whose graph is a straight line.

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Standard Form

• Ax By C where A, B, C are real numbers with A
  and B not both zero. If A, B, C are integers,
  the equation is in standard form.

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• Example
• Are these equations in standard form?
• 2x 5y 7
• 0.5x 4y 12
• x2y 3y 4
• 1/x 3y 1

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Graph the following lines

• 2x 3y 6
• x -2
• y 3

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SECTION 8-3

• The Slope of a Line

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Property

• A basic property of a straight line is that its
  slope is constant.

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• SLOPE
• is the ratio of vertical change to the
  horizontal change. The variable m is used to
  represent slope.

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FORMULA FOR SLOPE

• m change in y-coordinate
• change in x-coordinate
• Or
• m rise
• run

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• SLOPE OF A LINE
• m y2 y1
• x2 x1

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• Find the slope of the line that contains the
  given points.
• M(4, -6) and N(-2, 3)

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• Find the slope of the line that contains the
  given points.
• M(-2, 3) and N(4, 8)

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• HORIZONTAL LINE
• a horizontal line containing the point
• (a, b) is described by the equation y b and has
  slope of 0

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• VERTICAL LINE
• a vertical line containing the point (c, d) is
  described by the equation x c and has no slope

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SECTION 8-4

• The Slope Intercept Form of a Linear Equation

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• SLOPE-INTERCEPT FORM
• y mx b
• where m is the slope and b is the y -intercept

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• Y-Intercept
• is the point where the line intersects the y
  -axis.

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• X-Intercept
• is the point where the line intersects the
• x -axis.

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• Find the slope and y-intercept and use them to
  graph each equation
• 1. y -3/4x 6
• 2. 2x 5y 10

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THEOREM

• Let L1 and L2 be two different lines, with slopes
  m1 and m2 respectively.
• L1 and L2 are parallel if and only if m1m2

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THEOREM

• and
• L1 and L2 are perpendicular if and only if m1m2
  -1

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• Find the slope of a line parallel to the line
  containing points M and N.
• M(-2, 5) and N(0, -1)

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• Find the slope of a line perpendicular to the
  line containing points M and N.
• M(4, -1) and N(-5, -2)

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SECTION 8-5

• Determining an Equation of a Line

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Write an equation of a line with the given
y-intercept and slope

• m3 b 6
• Remember ymxb

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THEOREM

• Let P(x1,y1) be a point and m a real number.
  There is one line through P having slope m. An
  equation of the line is
• y y1 m (x x1)

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• Write an equation of a line with the given
  slope, passing through the given point.
• m 1/2 (-8, 4)

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Write an equation of a line passing through the
given points

• A(1, -3) B(3,2)

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SECTION 8-6

• Function Defined by Equations

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MAPPING DIAGRAM

• A picture showing a correspondence between two
  sets

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• MAPPING the relationship between the elements
  of the domain and range

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FUNCTION

• A correspondence between two sets, D and R, that
  assigns to each member of D exactly one member of
  R.

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• DOMAIN the set of all possible x-coordinates
• RANGE the set of all possible y-coordinates

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RANGE

• The set R of the function assigned to at least
  one member of D.

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SECTION 8-7

• Function Defined by Equations

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FUNCTION

• A correspondence between two sets, D and R, that
  assigns to each member of D exactly one member of
  R.

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• DOMAIN the set of all possible x-coordinates
• RANGE the set of all possible y-coordinates

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FUNCTIONAL NOTATION

• f (x) denotes the value of f at x

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ARROW NOTATION

• fx x 3
• Is read the function f that pairs x with x 3

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VALUES of a FUNCTION

• The members of its range.

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EXAMPLE 1

• Given f x?4x x2 with domain D 1,2,3,4,5
• Find the range of f.

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Example 2

• Given gx 4 3x x2 with domain D-1, 0,
  1, 2
• Find the range of g.

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SECTION 8-8

• Linear and Quadratic Functions

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Linear Function

• Is a function f that can be defined by f(x) mx
  b
• Where x, m and b are real numbers. The graph of
  f is the graph of y mx b, a line with slope m
  and y-intercept b.

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FUNCTION

• is a relation in which different ordered pairs
  have different first coordinates.

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RELATION

• Is any set of ordered pairs. The set of first
  coordinates in the ordered pairs is the domain of
  the relation, and

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RELATION

• and the set of second coordinates is the range.

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VERTICAL LINE TEST

• a relation is a function if and only if no
  vertical line intersects its graph more than once.

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Constant Function

• If f(x) mx b and
• m 0, then f(x) b for all x and its graph is
  a horizontal line y b

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Determine if Relation is a Function

• 2,1),(1,-2), (1,2)
• (x,y) ?x ? y 3

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SECTION 8-9

• Direct Variation

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SECTION 8-10

• Inverse Variation

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• END