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MAT091 :: Lecture Note :: Week 02
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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
0degree polynomial ( 91x^{0}, where x^{0} = 1
)91x^{2}  120x + 150
2degree polynomial x^{3} ÷ 91
3degree polynomial [ (1/91)x^{3}
]1 ÷ (x + 2)
not a polynomial (variable in denominator) x^{2} + 2x  1
not a polynomial (negative exponent) x^{4} + x^{3} + x^{1}
not a polynomial (negative exponent) 91 ÷ x^{4}
4degree polynomial 91x^{(1/2)}
not a polynomial (fractional exponent) 5 + x + x^{2} + x^{5}
5degree 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 ===================== 0degree constant 1degree linear 2degree quadratic 3degree cubic 4degree quartic 5degree quinticA polynomial with one term is called a monomial, two terms a binomial, and three terms a trinomial.
91 (0degree monomial) 91 + 29 (0degree binomial) x + 1 (1degree binomial) x^2 + 2 (2degree binomial) x^2 + x + 1 (2degree trinomial) x^3  1 (3degree binomial) x^3 + x^2 + x + 1 (3degree quadnomial?)The standard form for a polynomial is as follows.
a_{n}x^{n} + a_{n1}x^{n1} + ... a_{2}x^{2} + a_{1}x + a_{0}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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Solving equations can sometimes be made "easier" if polynomials are simplified. One step in simplifying polynomials is the "combining" of liketerms. Liketerms are terms having the same variables raised to the same exponents.
5x + 2x = 7x 2x  5x = 2x + (5x) = 3x x^{2} + 4x^{2} = 5x^{2} [x^{2} = 1x^{2}] 9x^{10} + 9y^{10} are not liketerms [different variables] 5x^{2} and 2x are not liketerms [different exponents]AlgebraHelp.com:: Equation Simplifying Calculator [opens new window]
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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 singlecharacter letters. For example,
3n
is3 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 = 8The 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 = 26Again, 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 coefficientThe 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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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 42(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 valuesb. 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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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 +5The 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 +  vRight 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
xaxis
and the vertical line is theyaxis
. < verticalaxis (yaxis)    o < horizonalaxis (xaxis) \  \  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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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 nameThe 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 8Public domain dotpng 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.
Related BABs and External Hyperlink(s)
 MathBABs.us:: Collection of Algebra BABs
 Wikipedia.org:: Function (mathematics)
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A constant function is a function that produces the same output regardless of the input.
f(x) = k, where k is a constantThe graph of a constant function is a horizontal line.
If zero is in the function's domain, then the constant function intersects the verticalaxis 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 verticalaxis at the point (0,3).  5  ABC A: (2,3) B: (0,3) C: (2,3)   + 2  2     5Unless 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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A identity function is a function that returns its input for all valid inputs.
f(x) = xThe 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 verticalaxis 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 verticalaxis 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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The absolute value function outputs the absolute value of its input.
f(x) = xThe 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     5If 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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