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MAT091 :: Lecture Note :: Week 02
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Define: Polynomial

The following linear function definition was copied from the Wikipedia.

   "A linear function is a first degree polynomial function of 
    one variable. These functions are called 'linear' because 
    they are precisely the functions whose graph in the Cartesian 
    coordinate plane is a straight line."

Understanding the definition for a linear function requires knowledge of polynomials and their respective degrees.

A polynomial is an expression that is "constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents."

The degree of a polynomial is the "largest exponent on any of the variables contained in the expression."

91 0-degree polynomial (91x0, where x0 = 1)
91x2 - 120x + 150 2-degree polynomial
x3 ÷ 91 3-degree polynomial [(1/91)x3]
1 ÷ (x + 2) not a polynomial (variable in denominator)
x-2 + 2x - 1 not a polynomial (negative exponent)
x4 + x3 + x-1 not a polynomial (negative exponent)
91 ÷ x-4 4-degree polynomial
91x(1/2) not a polynomial (fractional exponent)
5 + x + x2 + x5 5-degree polynomial

Polynomials are built from terms called monomials, which consist of a constant (called a coefficient) multiplied by "one or more variables. Each variable may have a constant positive whole number exponent."

A zero degree polynomial is call constant. A first degree polynomial is called linear. A second degree polynomial is called a quadratic, and a polynomial of degree three is called cubic.

   polynomial   name
   =====================
   0-degree     constant
   1-degree     linear
   2-degree     quadratic
   3-degree     cubic
   4-degree     quartic
   5-degree     quintic

A polynomial with one term is called a monomial, two terms a binomial, and three terms a trinomial.

   91                   (0-degree monomial)
   91 + 29              (0-degree binomial)
   x + 1                (1-degree binomial)
   x^2 + 2              (2-degree binomial)
   x^2 + x + 1          (2-degree trinomial)
   x^3 - 1              (3-degree binomial)
   x^3 + x^2 + x + 1    (3-degree quadnomial?)

The standard form for a polynomial is as follows.

   anxn + an-1xn-1 + ...  a2x2 + a1x + a0

The "Fundamental Theorem of Algebra" states that all polynomials can be factored, but the theorem is not covered in beginning Algebra courses.

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Like Terms

Solving equations can sometimes be made "easier" if polynomials are simplified. One step in simplifying polynomials is the "combining" of like-terms. Like-terms are terms having the same variables raised to the same exponents.

   5x + 2x = 7x
   2x - 5x = 2x + (-5x) = -3x
   x2 + 4x2 = 5x2 [x2 = 1x2]
   9x10 + 9y10 are not like-terms [different variables]
   5x2 and 2x are not like-terms [different exponents]

AlgebraHelp.com:: Equation Simplifying Calculator [opens new window]

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Algebraic Expressions

An algebraic expression is a mathematical expresssion that contains at least one variable along with zero or more numbers (i.e. constants) and zero or more arithmetic operators.

The following are algebraic expressions.

   n
   3n
   5n + 10
   2x^2 - 2y^3
   9a + 2b - 3c
   11(x + y^-3 + 2w)

Variables are unknown values that are represented by single-character letters. For example, 3n is 3 times 'n', where 'n' is a variable. If a value is assigned to 'n', then the expression can be evaluated.

   3n + 8

   if n = 3, then 3(3) + 8 = 100
   if n = -5, then 3(-5) + 8 = -7
   if n = 0, then 3(0) + 8 = 8

The replacement (or swapping) of variables with values can be described in a variety of ways: Values can be "plugged into" or "assigned to" variables; or variables can be "replaced with" or "substituted with" values; or we "let" variables be certain values.

   5x + 1  assign 5 to x ........... 5(5) + 1 = 25 + 1 = 26
   5x + 1  replace x with 5 ........ 5(5) + 1 = 25 + 1 = 26
   5x + 1  plug 5 into x ........... 5(5) + 1 = 25 + 1 = 26
   5x + 1  substitute x with 5 ..... 5(5) + 1 = 25 + 1 = 26
   5x + 1  let x be 5 .............. 5(5) + 1 = 25 + 1 = 26

Again, expressions are typically evaluated after variables have been assigned (i.e. given) values.

If a variable is multiplied by a constant, then the constant is called a coefficient.

   3n ...... 'n' is a variable and 3 is a coefficient
   x ....... 'x' is a variable and 1 is the coefficient
   -1y ..... 'y' is a variable and -1 is the coefficient

The combination of a coefficent and variable is called a factor. Factors are numbers, variables, and expressions that are multiplied together to produce a product.

Factors separated by addition and subtraction operators are called terms.

   3n + 1 .............. 1 factor, 2 terms
   4a + 2b - 10 ........ 2 factors (4a and 2b), 3 terms
   7x - 5 + 8y - 11 .... 2 factors (7x and 8y), 4 terms
   2x^3 + 10^2.......... 1 factor (2x^3), 2 terms
   3(4) - 2(1 + 3)...... 2 factors, 2 terms

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Simplifying Expressions

   a.  Simplify:  4 - 2(x + 3)
       4 + -2(x + 3)     # rewrite a - b as a + -b
       4 + -2x + -6      # distribute -2
       -2x - 2           # combine like terms (the constants)
       -2x + -2          # rewrite as an addition problem
       -2(x + 1)         # factor out a -2

       check work:  4 - 2(x + 3)               -2(x + 1)

       let x = 0... 4 - 2(0 + 3)               -2(0 + 1)
                    4 - 2(3) = 4 - 6 = -2      -2(1) = -2

       let x = 1... 4 - 2(1 + 3)               -2(1 + 1)
                    4 - 2(4) = 4 - 8 = -4      -2(2) = -4

       let x = -1... 4 - 2(-1 + 3)             -2(-1 + 1)
                     4 - 2(2) = 4 - 4 = 0      -2(0) = 0

       check work using the calculator:
         press Y=
         enter 4-2(x+3) into y1
         enter -2(x+1) into y2
         press GRAPH to see one line
         press 2nd GRAPH to see table values
         press Y=
         change y2 to -2(x+2)
         press GRAPH to see two lines
         press 2nd GRAPH and compare table values

   b.  Simplify:  (x^2 + x - 5) - (2x^2 - x + 3)
       # convert subtractions into additions
       (x^2 + x + -5) + (-1)(2x^2 + -x + 3)   

       x^2 + x + -5 + -2x^2 + x + -3      # drop ()s and distribute -1
       x^2 + -2x^2 + x + x + -5 + -3      # combine like terms (step 1)
       -x^2 + 2x + -8                     # combine like terms (step 2)

       check work:  let x = 0
       (x^2 + x - 5) - (2x^2 - x + 3)
       (0^2 + 0 - 5) - (2(0^2) - 0 + 3) = -5 - 3 = -8

       -x^2 + 2x + -8
       -0^2 + 2(0) + -8 = 0 + 0 + -8 = -8

       check work:  let x = 1
       (x^2 + x - 5) - (2x^2 - x + 3)
       (1^2 + 1 - 5) - (2(1^2) - 1 + 3) 
       (     -3    ) - (      4       ) = -7

       -x^2 + 2x + -8
       -1^2 + 2(1) + -8 
        -1  + 2    + -8 = -7

       check work:  let x = -1
       (x^2 + x - 5) - (2x^2 - x + 3)
       ((-1)^2 + -1 - 5) - (2(-1^2) - -1 + 3)
       (   1   + -1 - 5) - (2       - -1 + 3)
       (       -5      ) - (     6          ) = -11

       -x^2 + 2x + -8
       -(-1)^2 + 2(-1) + -8 = -1 + -2 + -8 = -11

       check work using the calculator:
          press Y=
          enter (x^2 + x - 5) - (2x^2 - x + 3) into y1
          enter -x^2 + 2x + -8  into y2
          on the y2 line... left arrow to the \ and press ENTER
          press GRAPH to see only one parabola
          press 2nd GRAPH to compare table values

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Define: horizontal and vertical

The horizon is the "apparent junction of earth and sky."

The following is a number line written horizontally.

   <--+---+---+---+---+---+---+---+---+---+---+-->
         -5  -4  -3  -2  -1   0  +1  +2  +3  +4  +5

The vertical is "perpendicular to the plane of the horizon or to a primary axis." [Perpendicular is "being at right angles to a given line or plane."]

The following is a number line written vertically.

      ^
      |
   +5 +
      |
   +4 +
      |
   +3 +
      |
   +2 +
      |
   +1 +
      |
    0 +
      |
   -1 +
      |
   -2 +
      |
   -3 +
      |
   -4 +
      |
   -5 +
      |
      v
Right Angle

Perpendicular lines intersect and their point of intersection is called the origin. Perpendicular lines form four right angles at the origin. A right angle measures ninety degrees (90°).

With respect the Cartesian coordinate system, the horizonal line is the x-axis and the vertical line is the y-axis.

           | <-- vertical-axis (y-axis)
           |
           |
           |
   --------o-------- <-- horizonal-axis (x-axis)
           |\
           | \
           |  o is the origin
           |   

Note: The Cartesian coordinate system was developed by French mathematician and philosopher Rene Descartes in approximately 1637.

RoadHacker:: Horizon/Vertical Slideshow [opens new window]

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Introduction to Functions

A function is a "relationship" that maps each input into one and only one output.

"Relationship" is quoted because functions can be defined in a variety of ways. The following was copied from the Wikipedia.

   "The mathematical concept of a function expresses the 
    intuitive idea of deterministic dependence between two 
    quantities, one of which is viewed as primary (the independent 
    variable, argument of the function, or its 'input') and the 
    other as secondary (the value of the function, or 'output'). 

In the Wikipedia's function definition, the word deterministic implies that every input to a function always results in the same output.

Terminology

The valid inputs to a function represents the domain of the function.

The range of a function are all the possible outputs.

Inputs are independent of the outputs, but the outputs are dependent on the inputs.

Functions are typically given one letter names. In many instances, inputs are labeled (named) 'x' and outputs are labeled (named) 'y'.

              +----------+
   input ---> | function | ---> output
     x        +----------+         y

   y = f(x)   or    f(x) = y

   y is the output, x is the input, f() is the function name

The following is a function with an implementation.

   f(x) = 2x                   [each input is multiplied by 2]

   f(-2) outputs -4            [input is -2]
   f(-1) outputs -2            [input is -1]
   f(0) outputs 0              [input is 0]
   f(1) outputs 2              [input is 1]
   f(2) outputs 4              [input is 2]

   f() is a function because each input produces only one output.
   Notice how each output value depends on the input value.

   The domain of f() is all real numbers.
   The range of f() is all real numbers.

   Function f() could be named doubler() because the output
   is always double (or 2 times) the input.  

   doubler(5) outputs 10
   doubler(-10) outputs -20
   doubler(5 * 3) outputs 30
   doubler(1 - 2 - 3) outputs -8

Public domain dot-png from Wikipedia.org...

Are They Functions?
   Is f() a potential function?  If yes, what's its domain and range?

           +-------+
   1 ----> |  f()  | ---->  -1          f(1) = -1            (1, -1)
           +-------+
           +-------+
   2 ----> |  f()  | ---->  -2          f(2) = -2            (2, -2)
           +-------+
           +-------+
   3 ----> |  f()  | ---->  -3          f(3) = -3            (3, -3)
           +-------+


   Is g() a potential function?  If yes, what's its domain and range?

           +-------+
   1 ----> |  g()  | ---->  -1          g(1) = -1            (1, -1)
           +-------+
           +-------+
   2 ----> |  g()  | ---->  -2          g(2) = -2            (2, -2)
           +-------+
           +-------+
   3 ----> |  g()  | ---->  -2          g(3) = -2            (3, -2)
           +-------+

   Is h() a potential function?  If yes, what's its domain and range?

           +-------+
   1 ----> |  h()  | ---->  -1          h(1) = -1            (1, -1)
           +-------+
           +-------+
   2 ----> |  h()  | ---->  -2          h(2) = -2            (2, -2)
           +-------+
           +-------+
   2 ----> |  h()  | ---->  -3          h(2) = -3            (2, -3)
           +-------+
           +-------+
   3 ----> |  h()  | ---->  -4          h(3) = -4            (3, -4)
           +-------+

   Is Q() a potential function?  If yes, what's its domain and range?

           +-------+
   1 ----> |  Q()  | ---->  -1          Q(1) = -1            (1, -1)
           +-------+
           +-------+ 
   2 ----> |  Q()  | ---->  -2          Q(2) = -2            (2, -2)
           +-------+
           +-------+
   2 ----> |  Q()  | ---->  -2          Q(2) = -2            (2, -2)
           +-------+
           +-------+
   3 ----> |  Q()  | ---->  -3          Q(3) = -3            (3, -3)
           +-------+

If a function receives an input that is not in its domain, then the output of the function is "undefined." (i.e. A function does not work on inputs that are not in its domain.) If the domain of a function is not stated, then the its domain is all real numbers.

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Constant Function

A constant function is a function that produces the same output regardless of the input.

   f(x) = k, where k is a constant

The graph of a constant function is a horizontal line.

If zero is in the function's domain, then the constant function intersects the vertical-axis at its constant output value.

   f(x) = 3

   f(-2) = 3
   f(0) = 3
   f(2) = 3

   The graph for f() is a horizontal line that intersects 
   the vertical-axis at the point (0,3).

          | 5 
          |
   A------B------C        A: (-2,3)   B: (0,3)   C: (2,3)
          |
          |
   -------+-------
   -2     |      2
          |
          |
          |
          | -5

Unless stated otherwise, the domain of a constant function is all real numbers. The range of the constant function is the constant value that it always outputs.

Example of a Constant Functon

The following is the Speed At Stop Sign Function (SASSF).

   f(s) = 0 MPH

   's'  is the speed at which a vehicle approaches a stop sign
   f(s) is the speed a vehicle should be moving at a stop sign

   's' and f(s) are both in units of MPH (Miles Per Hour)

   domain:  0 < s < infinity
    range:  0

   note:  For this specific constant function, the output does 
          not depend on the input.

   note:  As of 8 June 2007, the land speed record is 763 MPH;
          therefore, the practical domain for this function 
          could be 0 < s < 763.
Constant Functions are Linear Functions

A constant function is a linear function having zero slope.

   f(x) = mx + b for a constant function is f(x) = 0x + b

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Identity Function

A identity function is a function that returns its input for all valid inputs.

   f(x) = x

The graph of an identity function is straight line having a slope of one.

If zero is in the function's domain, then the indentity function intersects the vertical-axis at point (0,0).

   f(x) = x              [implementation]

   f(-2) = -2
   f(-1) = -1
   f(0) = 0
   f(1) = 1
   f(2) = 2

   The graph for f() is a straight line having a slope of 
   one that intersects the vertical-axis at the point (0,0).

          | 5 
          |
          |                                              
          |     E             E: (2, 2)
   -2     |  D                D: (1, 1)
    ------C------             C: (0, 0)
       B  |     2             B: (-1, -1)
    A     |                   A: (-2, -2)
          |
          |
          | -5

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Absolute Value Function

The absolute value function outputs the absolute value of its input.

   f(x) = |x|

The graph of an absolute value function, where the domain is all real numbers, is two staight lines that intersect at point (0,0). There is a straight line in the 2nd quadrant that has a slope of -1 and there is a straight line in the 1st quadrant having a slope of 1.

   f(x) = |x|

   f(-4) = 4
   f(-2) = 2
   f(0) = 0
   f(2) = 2
   f(4) = 4

                | 5 
   A            |            E          A: (-4,4)    E: (4,4)
                |
         B      |      D                B: (-2,2)    D: (2,2)
                |
   -------------C-------------          C: (0,0)
  -4            |            4
                |
                |
                |
                | -5

If the domain of an absolute value function is all real numbers, then its range is all real numbers greater than or equal to zero.

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