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Mathematics > Relations & Functions > Flashcards

Relations & Functions Flashcards

1
Q

set

A

a well-defined collection of objects, which are called the elements of the set

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2
Q

empty set

A

∅ = {} = {x | x ≠ x}

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3
Q

set of natural numbers

A

N = {1, 2, 3, …}

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4
Q

set of whole numbers

A

W = {0, 1, 2, …}

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5
Q

set of integers

A

Z = {…, -3, -2, -1, 0, 1, 2, 3, …}

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6
Q

set of rational numbers

A

Q = {a/b | a ∈ Z and b ∈ Z}

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7
Q

set of real numbers

A

R = {x | x possesses a decimal representation}

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8
Q

set of irrational numbers

A

P = {x | x is a non-rational real number—that is, a decimal that does not repeat or terminate}

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9
Q

set of complex numbers

A

C = {a + bi | a,b ∈ R and i = √-1}

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10
Q

interval

A

the set of all real numbers lying between two fixed endpoints with no gaps

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11
Q

intersection of two sets

A

A ∩ B = {x | x ∈ A and x ∈ B}

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12
Q

union of two sets

A

A ∪ B = {x | x ∈ A or x ∈ B (or both}

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13
Q

Quadrant I

A

x > 0, y > 0

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14
Q

Quadrant II

A

x < 0, y > 0

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15
Q

Quadrant III

A

x < 0, y < 0

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16
Q

Quadrant IV

A

x > 0, y < 0

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17
Q

(a, b) and (c, d) are symmetric about the x-axis if

A

a = c and b = -d

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18
Q

(a, b) and (c, d) are symmetric about the y-axis if

A

a = -c and b = d

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19
Q

(a, b) and (c, d) are symmetric about the origin if

A

a = -c and b = -d

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20
Q

to reflect (x, y) about the x-axis,

A

replace y with -y

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21
Q

to reflect (x, y) about the y-axis,

A

replace x with -x

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22
Q

to reflect (x, y) about the origin,

A

replace x with -x and y with -y

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23
Q

the distance between points P(a, b) and Q(c, d) is

A

d = √[(c – a)² + (d – b)²]

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24
Q

the midpoint M of the line segment connecting P(a, b) and Q(c, d) is

A

M = [(a + c)/2, (b + d)/2]

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25
Q

relation

A

a set of points in the plane

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26
Q

the graph of x = a is

A

a vertical line through (a, 0)

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27
Q

the graph of y = b is

A

a horizontal line through (0, b)

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28
Q

the fundamental graphing principle

A

The graph of an equation is the set of points which satisfy the equation. That is, a point (x, y) is on the graph if and only if x and y satisfy the equation.

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29
Q

x-intercept

A

a point on a graph which is also on the x-axis

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30
Q

y-intercept

A

a point on the graph which is also on the y-axis

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31
Q

how to find an x-intercept

A

set y = 0 and solve for x to find the x-intercepts (x, 0)

32
Q

how to find a y-intercept

A

set x = 0 and solve for y to find the y-intercepts (0, y)

33
Q

test the graph of an equation for symmetry about the y-axis

A

substitute (–x, y) and simplify. The graph is symmetric about the y-axis if the result is equivalent to the original equation

34
Q

test the graph of an equation for symmetry about the x-axis

A

substitute (x, –y) and simplify. The graph is symmetric about the x-axis if the result is equivalent to the original equation

35
Q

test the graph of an equation for symmetry about the origin

A

substitute (–x, –y) and simplify. The graph is symmetric about the origin if the result is equivalent to the original equation

36
Q

function (definition)

A

a rule that associates a unique output with each input

37
Q

the vertical line test

A

a set of points in the plane represents y as a function of x if and only if no two points lie on the same vertical line

38
Q

domain (definition)

A

the set of all allowable inputs (x-values) of a function

39
Q

range (definition)

A

the set of outputs (y-values) that result when x varies over the domain of a function

40
Q

how to determine whether an equation represents y as a function of x

A

solve for y and determine whether each choice of x will determine only one corresponding value of y

41
Q

independent variable

A

x is the independent variable (or argument) of f

42
Q

dependent variable

A

y is the dependent variable of f

43
Q

difference quotient

A

[f(x+h) – f(x)]/h

44
Q

the fundamental graphing principle for functions

A

The graph of a function f is the set of points which satisfy the equation y = f(x). That is, the point (x, y) is on the graph of f if and only if y = f(x).

45
Q

zeros (definition)

A

the values of x for which f(x) = 0

the x-coordinates of the points where the graph of f intersects the x-axis

also called roots or x-intercepts

46
Q

the graph of a function f is symmetric about the y-axis

A

if and only if f(–x) = f(x) for all x in the domain of f

47
Q

the graph of a function f is symmetric about the origin

A

if and only if f(–x) = –f(x) for all x in the domain of f

48
Q

even function

A

a function whose graph is symmetric about the y-axis

49
Q

odd function

A

a function whose graph is symmetric about the origin

50
Q

a function f is increasing on an interval I if

A

f(a) < f(b) for all real numbers a, b in I with a < b

51
Q

a function f is decreasing on an interval I if and only if

A

f(a) > f(b) for all real numbers a, b in I with a < b

52
Q

a function f is constant on an interval I if and only if

A

f(a) = f(b) for all real numbers a, b in I

53
Q

Suppose f is a function with f(a) = b. f has a local maximum at (a, b) if and only if

A

there is an open interval I containing a for which f(a) ≥ f(x) for all x in I

54
Q

Suppose f is a function with f(a) = b. f has a local minimum at the point (a, b) if and only if

A

there is an open interval I containing a for which f(a) ≤ f(x) for all x in I

f(a) = b is a local minimum value of f in this case

55
Q

Suppose f(a) = b. The value b is a maximum of f if

A

b ≥ f(x) for all x in the domain of f

56
Q

Suppose f(a) = b. The value b is called the minimum of f if

A

b ≤ f(x) for all x in the domain of f

57
Q

Suppose k is a positive number. To graph y = f(x) + k,

A

shift the graph up k units by adding k to the y-coordinates of the points on the graph of f

58
Q

Suppose k is a positive number. To graph y = f(x) – k,

A

shift the graph of y = f(x) down k units by subtracting k from the y-coordinates of the points on the graph of f

59
Q

how to shift a function vertically

A

adding to or subtracting from the output of a function causes the graph to shift up or down, respectively

add k to the y-coordinates to shift f up k units: y = f(x) + k

subtract k from the y-coordinates to shift y down k units: y = f(x) – k

60
Q

Suppose h is a positive number. To graph y = f(x + h),

A

shift the graph of y = f(x) left h units by subtracting h from the x-coordinates of the points on the graph of f

61
Q

Suppose h is a positive number. To graph y = f(x – h),

A

shift the graph of y = f(x) right h units by adding h to the x-coordinates of the points on the graph of f

62
Q

how to shift a graph horizontally

A

adding to or subtracting from the input of a function amounts to shifting the graph left or right, respectively

63
Q

To graph y = –f(x),

A

reflect the graph of y = f(x) across the x-axis by multiplying the y-coordinates of the points on the graph of f by -1

64
Q

To graph y = f(–x),

A

reflect the graph of y = f(x) across the y-axis by multiplying the x-coordinates of the points on the graph of f by –1

65
Q

how to reflect a graph across the x-axis

A

multiply the output from a function [the y-coordinates] by –1

66
Q

how to reflect a graph across the y-axis

A

multiply the input [the x-coordinates] to the function by -1

67
Q

rigid transformations (definition)

A

vertical shifts, horizontal shifts, and reflections

these transformations change the position and orientation of a graph in the plane, but not its shape

68
Q

non-rigid transformations (definition)

A

transformations, including vertical and horizontal scalings, that change the shape of the graph

69
Q

Suppose a > 0. To graph y = af(x),

A

multiply all of the y-coordinates of the points on the graph of f by a

70
Q

how to stretch a graph vertically

A

multiply all of the y-coordinates of the points by a factor of a, a > 1

71
Q

how to shrink a graph vertically

A

multiply all of the y coordinates on the points of the graph by a, 0 < a < 1

72
Q

Suppose b > 0. To graph y = f(bx),

A

divide all of the x-coordinates of the points on the graph of f by b

we say that f has been horizontally scaled by a factor of 1/b

73
Q

how to stretch a graph horizontally

A

divide all of the x-coordinates of the points of the graph by b, 0 < b < 1

y = f(bx), f stretches horizontally by a factor of 1/b

74
Q

how to shrink a graph horizontally

A

divide all of the x-coordinates of the points on the graph of f by b, b > 1

if b > 1, y = f(bx) and f shrinks horizontally by a factor of b

75
Q

how to graph g(x) = Af(Bx + H) + K

A
  1. subtract H from each x-coordinate, shifting f to the left if H > 0 or right if H < 0
  2. divide the (new) x-coordinates by B, resulting in a horizontal scaling (and a reflection about the y-axis if B > 0)
  3. multiply the y-coordinates by A, resulting in a vertical scaling (and a reflection about the x-axis if A < 0)
  4. add K to each of the (new) y-coordinates, resulting in a vertical shift up if K > 0 or down if K < 0
76
Q

real-valued function of a real variable (definition)

A

a function where the independent and dependent variables are real numbers

77
Q

natural domain (definition)

A

where the domain for a real-valued function of a real variable is not explicitly stated, then the natural domain consists of all real numbers for which the formula yields a real value

Mathematics (4 decks)

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