Functions

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1.1

• Functions

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Quick Review

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What youll learn about

• Numeric Models
• Algebraic Models
• Graphic Models
• The Zero Factor Property
• Problem Solving
• Grapher Failure and Hidden Behavior
• A Word About Proof
• and why
• Numerical, algebraic, and graphical models
  provide different
• methods to visualize, analyze, and understand
  data.

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Mathematical Model

• A mathematical model is a mathematical
• structure that approximates phenomena for the
• purpose of studying or predicting their behavior.

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Numeric Model

• A numeric model is a kind of mathematical
• model in which numbers (or data) are analyzed to
  gain insights into phenomena.

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Algebraic Model

• An algebraic model uses formulas to relate
• variable quantities associated with the
• phenomena being studied.

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Example Comparing Pizzas

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Example Comparing Pizzas

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Graphical Model

• A graphical model is a visible representation of
• a numerical model or an algebraic model that
• gives insight into the relationships between
• variable quantities.

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Example Solving an Equation

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Example Solving an Equation

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Fundamental Connection

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Pólyas Four Problem-Solving Steps

1. Understand the problem.
2. Devise a plan.
3. Carry out the plan.
4. Look back.

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A Problem-Solving Process

• Step 1 Understand the problem.
• Read the problem as stated, several times if
  necessary.
• Be sure you understand the meaning of each term
  used.
• Restate the problem in your own words. Discuss
  the problem with others if you can.
• Identify clearly the information that you need to
  solve the problem.
• Find the information you need from the given
  data.

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A Problem-Solving Process

• Step 2 Develop a mathematical model of the
  problem.
• Draw a picture to visualize the problem
  situation. It usually helps.
• Introduce a variable to represent the quantity
  you seek.
• Use the statement of the problem to find an
  equation or inequality that relates the variables
  you seek to quantities that you know.

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A Problem-Solving Process

• Step 3 Solve the mathematical model and support
  or confirm the solution.
• Solve algebraically using traditional algebraic
  models and support graphically or support
  numerically using a graphing utility.
• Solve graphically or numerically using a graphing
  utility and confirm algebraically using
  traditional algebraic methods.
• Solve graphically or numerically because there is
  no other way possible.

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A Problem-Solving Process

• Step 4 Interpret the solution in the problem
  setting.
• Translate your mathematical result into the
  problem setting and decide whether the result
  makes sense.

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Example Seeing Grapher Failure

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Example Seeing Grapher Failure

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1.1(a)/1.2

• Functions and Their Properties

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Quick Review

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What youll learn about

• Function Definition and Notation
• Domain and Range
• Continuity
• Increasing and Decreasing Functions
• Boundedness
• Local and Absolute Extrema
• Symmetry
• Asymptotes
• End Behavior
• and why
• Functions and graphs form the basis for
  understanding
• The mathematics and applications you will see
  both in your work
• place and in coursework in college.

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Function, Domain, and Range

• 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.

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Mapping
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Example Seeing a Function Graphically

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Example Seeing a Function Graphically
The graph in (c) is not the graph of a function.
There are three
points on the graph with x-coordinates 0.
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Vertical Line Test

• A graph (set of points (x,y)) in the xy-plane
• defines y as a function of x if and only if no
• vertical line intersects the graph in more than
  one
• point.

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Agreement

• Unless we are dealing with a model that
• necessitates a restricted domain, we will assume
  that the domain of a function defined by an
  algebraic expression is the same as the domain of
  the algebraic expression, the implied domain.
• For models, we will use a domain that fits the
  situation, the relevant domain.

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Example Finding the Domain of a Function

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Example Finding the Domain of a Function

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Example Finding the Range of a Function

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Example Finding the Range of a Function

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Continuity
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Example Identifying Points of Discontinuity
Which of the following figures shows functions
that are discontinuous at x 2?

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Example Identifying Points of Discontinuity

• Which of the following figures shows functions
  that are
• discontinuous at x 2?

The function on the right is not defined at x 2
and can not be continuous there. This is a
removable discontinuity.
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Increasing and Decreasing Functions
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Increasing, Decreasing, and Constant Function on
an Interval

• A function f is increasing on an interval if, for
  any two points in the interval, a positive change
  in x results in a positive change in f(x).
• A function f is decreasing on an interval if, for
  any two points in the interval, a positive change
  in x results in a negative change in f(x).
• A function f is constant on an interval if, for
  any two points in the interval, a positive change
  in x results in a zero change in f(x).

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Example Analyzing a Function for
Increasing-Decreasing Behavior

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Example Analyzing a Function for
Increasing-Decreasing Behavior

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Lower Bound, Upper Bound and Bounded

• A function f is bounded below of 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.
• A function f is bounded above of 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 a upper bound of f.
• A function f is bounded if it is bounded both
  above and below.

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Local and Absolute Extrema

• A local maximum of a function f is a value
  f(c) that is greater than or equal to all range
  values of f on some open interval containing c.
  If f(c) is greater than or equal to all range
  values of f, then f(c) is the maximum (or
  absolute maximum) value of f.
• A local minimum of a function f is a value
  f(c) that is less than or equal to all range
  values of f on some open interval containing c.
  If f(c) is less than or equal to all range values
  of f, then f(c) is the minimum (or absolute
  minimum) value of f.
• Local extrema are also called relative
  extrema.

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Example Identifying Local Extrema

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Example Identifying Local Extrema

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Symmetry with respect to the y-axis
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Symmetry with respect to the x-axis
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Symmetry with respect to the origin
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Example Checking Functions for Symmetry

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Example Checking Functions for Symmetry

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Horizontal and Vertical Asymptotes