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MAT122 :: Lecture Note :: Week 02
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BAB::The
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Quote of the Week
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Learning is what most adults will do for a living in the 21st century.
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Grigori Perelman
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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 in 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 x 3) outputs 30 doubler(1 - 2 - 3) outputs -8Are They Functions?
Is f() a potential function? If yes, what's its domain and range? +-------+ 1 ----> | f() | ----> -1 +-------+ +-------+ 2 ----> | f() | ----> -2 +-------+ +-------+ 3 ----> | f() | ----> -3 +-------+ Is g() a potential function? If yes, what's its domain and range? +-------+ 1 ----> | g() | ----> -1 +-------+ +-------+ 2 ----> | g() | ----> -2 +-------+ +-------+ 3 ----> | g() | ----> -2 +-------+ Is h() a potential function? If yes, what's its domain and range? +-------+ 1 ----> | h() | ----> -1 +-------+ +-------+ 2 ----> | h() | ----> -2 +-------+ +-------+ 2 ----> | h() | ----> -3 +-------+ +-------+ 3 ----> | h() | ----> -4 +-------+ Is i() a potential function? If yes, what's its domain and range? +-------+ 1 ----> | i() | ----> -1 +-------+ +-------+ 2 ----> | i() | ----> -2 +-------+ +-------+ 2 ----> | i() | ----> -2 +-------+ +-------+ 3 ----> | i() | ----> -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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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 domain of a function are all the inputs for which a function is defined.
unknown functions and their respective domains ============================================== f(x) -- all real numbers (-∞, ∞) R: -∞ < x < ∞ g(x) -- all integers (-∞, ∞) Z: -∞ < x < ∞ h(x) -- all positive integers including zero [0, ∞) Z: x ≥ 0 i(x) -- all real numbers between 0 and 1, exclusive (0,1) R: 0 < x < 1 j(x) -- all real numbers between 0 and 1, inclusive [0,1] R: 0 ≤ x ≤ 1 k(x) -- all negative real numbers (-∞ 0) R: x < 0 l(x) -- all positive integers less than or equal to 10 (0, 10] Z: 0 < x ≤ 10 R implies real number Z implies integerDomain values are the indepedent values.
The range of a function are all the outputs produced by the fuction.
Range values are the dependent values. The output of a function depends on its input; therefore, the range of a function depends on its domain.
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Functions are sometimes categorized as either increasing or decreasing.
An increasing function is a function whose output values increase (i.e. get larger) when the inputs increase.
f(0) = 5 f(1) = 10 f(2) = 15 f(3) = 20 f(x) appears to be an increasing functionA decreasing function is a function whose output values decrease (i.e. get smaller) when the inputs increase.
g(0) = 100 g(1) = 90 g(2) = 80 g(3) = 70 g(x) appears to be a decreasing functionA constant function
f(x) = k(where 'k' is a constant) is neither increasing nor decreasing.The identity function
f(x) = xis an increasing function.Some functions can be both increasing and decreasing. For example, over an interval A, a function might be increasing, while it decreases over an interval B. Review the absolute value function.
abs(x) is decreasing from (-INF, 0) abs(x) is zero at 0 abs(x) is increasing from (0, INF)Technical definitions for increasing and decreasing functions.
increasing function: ==================== Function f(x) increases on an interval I if f(b) > f(a) for all b > a, where a,b are in I. decreasing function: ==================== Function f(x) decreases on an interval I if f(b) < f(a) for all b > a, where a,b are in I.
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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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M-W.com defines linear as follows.
"of, relating to, resembling, or having a graph that is a line and especially a straight line" [...and...] "having or being a response or output that is directly proportional to the input"Observe that the word linear contains the word line.
A linear function is a 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."
The following are equations for a line.
slope-intercept form: y = mx + b ... used when slope and y-intercept are known point-slope form: y - y1 = m(x - x1) ... where (x1, y1) is a point on the line having slope m standard form: Ax + By = CSlope is another way of saying "rate of change" and linear functions have a constant rate of change.
Slope is often described as the ratio "rise over run."
Linear functions with a positive constant rate of change are increasing functions. Linear functions with a negative constant rate of change are decreasing functions.
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Memorize the following
slope formula: m = (y2 - y1) / (x2 - x1) slope-intercept form: y = mx + b point-slope form: y - y1 = m(x - x1)
[exercise]Given the two points(1, 5)and(3, 15), find the equation of the line.step 1: calculate the slope m (15 - 5) / (3 - 1) = 10 / 2 = 5 step 2: substitute one of the points and the slope into the point-slope form equation for a line y - 5 = 5(x - 1) # point (1, 5) selected; m=5 y1=5, x1=1 step 3: solve for y y - 5 = 5x - 5 # distributive property on right-side y = 5x - 5 + 5 # add 5 to both sides y = 5x + 0 # this is slope-intercept formSince the 'b' value is zero we know that this line passes through point
(0, 0). Recall, the 'b' value is the y-intercept (vertical-intercept) and it is obtained by setting the input to zero.Solving this problem using the TI-83 calcuator.
press ON press STAT press 1:EDIT if there are values in L1, then left arrow to L1 column, if necessary up arrow to L1 cell, if necessary press CLEAR press ENTER if there are values in L2, then up arrow to L1 cell right arrow to L2 cell press CLEAR press ENTER if necessary, use arrow keys to get to line 1 of L1 1 ENTER 3 ENTER right arrow 5 ENTER 15 ENTER There are now two point entered. L1 | L2 -------------- 1 | 5 3 | 15 press STAT right arrow to CALC press 4:LinReg(ax + b) home screen says: LinReg(ax + b) [cursor] press 2ND 1 (L1) press COMMA (,) press 2ND 2 press COMMA (,) press VARS right arrow to Y-VARS press 1:Function press 1:Y1 home screen says: LinReg(ax + b) L1,L2,Y1 [cursor] press ENTER home screen says: LinReg y = ax+b a = 5 b = 0 r^2 = 1 r = 1 press 2ND GRAPH to see a table press GRAPH to see a graph of the line press Y= to see the linear equation
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The slope of a line is a measure of its average rate of change (steepness?). The slope also indicates if the line is increasing (uphill?) or decreasing (downhill?). In linear equations, slope is represented by the letter m.
The following is how to find the slope between two unique points on a line.
point 1: (x1, y1) point 2: (x2, y2) slope = m = (y2 - y1) / (x2 - x2) note: x1 ≠ x2Slope represents the ratio of change-in-output over change-in-input.
m = Δy / Δx (change in output / change in input) Δ is the "delta" character Δy is read "change of" (Δy is read "change of" y) Δy = y2 - y1Slope is often described as the ratio "rise over run."
rise m = ------ runSome slope notes.
horizonal lines have slope 0 (y = f(x) = k, where k is a consant) vertical lines have undefined slope (x = k, where k is a consant) parallel lines have equal slopes (m1 = m2) perpendicular lines have negative inverse slopes (m2 = -1/m1)The following is a verbose form the slope-intercept equation.
output = slope * input + initial_value_if_any ...or... output = average_rate_of_change * input + initial_value_if_any
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Intercepts are points where lines intersect. In many cases, we need to know where lines intersect the horizonal- and vertical-axis (often called the x-intercept and y-intercept, respectively).
The vertical-intercept for a function is found by evaluating a function when its input is zero.
The horizontal-intercept for a function is found by setting its output to zero and finding what input value results in that output.
f(x) = 5(x) + 3 vertical-intercept (y-intercept)... set the input to 0 and evaluate: f(0) = 5(0) + 3 = 3 vertical-intercept is (0, 3) horizontal-intercept (x-intercept)... set the output to 0 and solve: 0 = 5(x) + 3 -3 = 5(x) -3/5 = x horizontal-intercept is (-3/5, 0)Generalizations.
given f(x) = m(x) + b vertical-intercept: (0, b) [notice input is 0] horizontal-intercept: (-b/m, 0) [notice output is 0]
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