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MAT092 :: Lecture Note :: Week 03
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[due: class 2 of week 3]
[due: class 2 of week 3]
Quote of the Week
{Furman.edu::
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As long as no one asks me, it seems to me that I know it;
but if someone asks me and I have to explain it,
it seems to me that I do not know it.
--
St. Augustine
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{philosopher and theologian;
more...}
[knowledge]
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BARS of the Week
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Marilyn Carlson, Michael Oehrtman and Patrick W. Thompson of Arizona State University have authored a 21 page dot-pdf titled: "Foundational Reasoning Abilities that Promote Coherence in Students' Understanding of Function." Their paper starts as follows.
"The concept of function is central to undergraduate mathematics, foundational to modern mathematics, and essential in related areas of the sciences." "Since 1888, there have been repeated calls for school curricula to place greater emphasis on functions."Let's look at yet another definition for a mathematical function.
"A relation for which each element of the domain corresponds to exactly one element of the range."Some function examples.
current world population is a function of time category of a hurricane is a function of wind speed cost of a loan is a function of its interest rate calories consumed is a function of serving size your current age is a function of your birthdate diameter of a circle is a function of its radiusBeginning algebra courses focus on function having one independent variable (i.e. one input), but there are many functions that depend on multiple inputs.
wind chill is a function of temperature & wind speed cost to gas a car is a function of #gallons pumped & cost per gallon perimeter of a rectangle is a function of its length & width duration of a road trip is a function of distance & avg. speed your genetic make-up is a function of your mom & dad total change collected is a function of #half-dollars, #quarters, #dimes, #nickels & #pennies
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A function takes one input value and produces one output value. The same input value always produces the same output value.
input domain indepedent horizontal-axis x-axis output range dependent vertical-axis y-axisIf a specific domain is not given for a function, then its domain is all real numbers.
If a function receives an input that is not in its domain, then the output of the function is undefined (i.e. the function doesn't work).
The output depends on the input. The input is independent of the output.
For a given domain, a function always produces the same range.
f(x) is read "f of x" [not f times x] g(n) is read "g of n" [not g times n] h(t) is read "h of t" [not h times t] f(x), g(n), h(t) all represent an output value f(x) is a function of x [f(x) depends on x] g(n) is a function of n [g(n) depends on n] h(t) is a function of t [h(t) depends on t]
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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 | | | | -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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The inputs and outputs of a function are often represented as ordered pairs (or points).
(input, output) ...or... (x, y) ...or... (x, f(x)) (t, n) ...or... (t, f(t)) (n, o) ...or... (n, f(n)) (a, b) ...or... (a, f(a))The input is always the first value recorded followed by its respective output.
In some instances, it is necessary to find the mid-point between two ordered-pairs.
mid-point of (a, b) and (c, d) is ((a+c)/2, (b+d)/2) mid-point of (2, 4) and (6, 12) is (4, 8) observe... ((2+6)/2, (4+12)/2) = (8/2, 16/2) = (4, 8)Data contained in tables can sometimes be represented as ordered-pairs.
input: month ......... output: #days month | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 ------------------------------------------------------------------ #days | 31 | 28 | 31 | 30 | 31 | 30 | 31 | 31 | 30 | 31 | 30 | 31 (1, 31) (2, 28) (3, 31) (4, 30) (5, 31) (6, 30) (7, 31) (8, 31) (9, 30) (10, 31) (11,30) (12, 31) #days depends on the month (i.e. the output depends on the input) The function f(m) takes a month as input and outputs the number of days in that month. f(m) = n f(1) = 31 f(2) = 28 f(7) = 31 f(12) = 31Ordered-pairs are points that can be graphed. The input is along the horizontal-axis and output is along the vertical-axis.
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The graph formed by the intersection of a horizontal line and a vertical line can be divided into four quadrants.
vertical-axis (y-axis; outputs) | 2 (II) | 1 (I) | -----------o----------- horizontal-axis (x-axis; inputs) | o is the origin (0, 0) 3 (III) | 4 (IV) |The top-right quadrant is always the first quadrant with the remining quadrants numbered in a counter-clockwise direction.
Quadrants are sometimes numbered (labeled) using Roman Numerals.
Ordered-pairs (i.e. points or coordinates) are plotted (graphed) as follows.
(+x, +y) ... quadrant 1 (I) (-x, +y) ... quadrant 2 (II) (-x, -y) ... quadrant 3 (III) (+x, -y) ... quadrant 4 (IV) ( 0, 0) ... origin ( 0, +y) ... vertical axis; above the origin ( 0, -y) ... vertical axis; below the origin (+x, 0) ... horizontal axis; right of origin (-x, 0) ... horizontal axis; left of origin
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When it comes to graphing functions, tables and other forms of data, input values are scaled along the horizontal-axis and output values are scaled along the vertical-axis.
When it comes to naming variables, the letter 'x' is often used to represent the input variable and the letter 'y' is used to represent the output variable; however, any letters can be used when it comes to naming input and output variables.
Table A =============================================================== input: year (y) | 2001 | 2002 | 2003 | 2004 | 2005 | 2007 --------------------------------------------------------------- output: # x-rays (x) | 55 43 81 24 43 61 The # of x-rays is a function of the year; i.e. year is the input and # of x-rays is the output. In this instance, the input variable is named 'y' and the output variable is named 'x'. Table A expressed using function notation. f(y) = x f(2001) = 55 f(2002) = 43 f(2003) = 81 ... f(2000) = undefined [cannot extrapolate] f(2006) = undefined [cannot interpolate]When input values are in units of time, then 't' is often used as the name for the input variable.
Table B ============================================================== input: time (t) | 1 | 2 | 3 | 4 | 5 | 6 | 7 -------------------------------------------------------------- output: # minutes (m) | 60 | 120 | 180 | 240 | 300 | 360 | 420 Time 't' is the input and it is in units of hours. One hour is 60 minutes, two hours is 120 minutes, etc. The number of minutes is a function of the time. Table B expressed using function notation. f(t) = m = 60(t) f(1) = 60 f(4) = 240 f(7) = 420 f(2.5) = 150 [interpolation] f(5.25) = 315 [interpolation] f(0) = 0 [extrapolation] f(8) = 480 [extrapolation] When graphed, 't' values (inputs) are scaled on the horizontal-axis and 'm' values (outputs) are scaled on the vertical-axis.Again, a mathematical function is a relationship (algorithm) that maps one input value to one and only one output value.
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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 single-character letters. For example,
3nis3 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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