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Calculus > Unit 1: Functions > Flashcards

Unit 1: Functions Flashcards

1
Q

Variables

A

Relationships among quantities

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2
Q

Function

A

A function f is a rule that assigns to each value x in a set D a unique value denoted f(x)

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3
Q

Domain

A

The set D of the function

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4
Q

Range

A

The set of all values of f(x) produced as x varies over the domain

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5
Q

Independent Variable

A

The variable associated with the domain

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6
Q

Dependent Variable

A

Belongs to the range

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7
Q

Graph

A

The set of all points (x,y) in the xy-plane that satisfy the equation y = f(x)

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8
Q

Argument

A

The expression on which the function works

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9
Q

Vertical Line Test

A

Every vertical line intersects the graph at most once, otherwise the graph is not a function

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10
Q

Relation

A

A set of points on a graph that does not correspond to a function

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11
Q

Composite Functions

A

Given two functions f and g, the composite function f • g is defined by (f • g) (x) = f(g(x))

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12
Q

Symmetric With Respect To The Y-Axis

A

If whenever the point (x,y) is on the graph, the point (-x,y) is also on the graph

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13
Q

Symmetric With Respect To The X-Axis

A

If whenever the point (x,y) is on the graph, the point (x,-y) is also on the graph

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14
Q

Symmetric With Respect To The Origin

A

If whenever the point (x,y) is on the graph, the point (-x,-y) is also on the graph

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15
Q

Even Function

A

Has the property that f(-x) = f(x) for all x in the domain

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16
Q

Odd Function

A

Has the property that f(-x) = -f(x) for all x in the domain

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17
Q

Polynomials

A

Have the form f(x) = anx^n + an-1x^n-1 + … + ax + a0, where the coefficients a0, a1,… are real numbers which cannot equal 0 and the nonnegative integer n is the degree of the polynomial. An nth degree polynomial can have as many n real zeros or roots-values of x at which f(x) = 0

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18
Q

Rational Functions

A

Are ratios of the form f(x) = p(x)/q(x), where p and q are polynomials

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19
Q

Algebraic Functions

A

Constructed using the operations of algebra

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20
Q

Exponential Functions

A

Have the form f(x) = b^x, where the base b cannot equal 1 is a positive real number

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21
Q

Logarithmic Functions

A

Have the form f(x) = logb x, where b > 0 and b cannot equal 1

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22
Q

Natural Logarithmic Function

A

Have the form f(x) = ln x, which also has the base b = e

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23
Q

Natural Exponential Function

A

Have the form f(x) = e^x, with base b = e, where e approx equal to 2.71828

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24
Q

Trigonometric Function

A

Are sin x, cos x, tan x, cot x, sec x, and csc x

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25
Q

Inverse Trigonometric Functions

A

The inverse of trigonometric functions

26
Q

Transcendental Functions

A

Consist of trigonometric, exponential, and logarithmic functions

27
Q

Technology

A

Graphing calculators and software used to produce graphs of functions

28
Q

Analytical Methods

A

Pencil-and-paper methods

29
Q

Linear Function

A

A straight-line graph that has the equation y = mx + b, where m and b are constants

30
Q

Piecewise Functions

A

Functions that have different definitions on different parts of the domain

31
Q

Piecewise Linear

A

If all the pieces of a piecewise function are linear

32
Q

Power Functions

A

Have the form f(x) = x^n, where n is a positive integer

33
Q

Root Functions

A

Have the form f(x) = x^1/n, where n > 1 is a positive integer

34
Q

Transformations

A

Given the real numbers a, b, c, and d and the function f, the graph of y = cf(a(x - b)) + d is obtained from the graph of y = f(x) in the following steps

y = f(x) —- horizontal scaling by a factor of abs value of a —- y = f(ax)

y = f(x) —- horizontal shift by b units —- y = f(a(x-b))

y = f(x) —- vertical scaling by a factor of abs value of c —- y = cf(a(x-b)

y = f(x) —- vertical shift by d units —- y = cf(a(x-b)) +d

35
Q

Properties of Exponential Functions

A
  1. Because b^x is defined for all real numbers, the domain of f is x: -infinity < x < infinity. Because b^x > 0 for all values of x, the range of f is y: 0 < y < infinity
  2. For all b > 0 b^0 = 1 and thus f(0) = 1
  3. If b > 1, then f is an increasing function of x.
  4. If 0 < b < 1, then f is a decreasing function of x
36
Q

The Natural Exponential Function

A

f(x) = e^x, which has the base e = 2.718281828459…

37
Q

Inverse Function

A

Given a function f, it’s inverse (if it exists) is a function f^-1 such that whenever y = f(x), then f^-1(y) = x

38
Q

One-To-One

A

A function f is one-to-one on a domain D if each value of f(x) corresponds to exactly one value of x in D.

39
Q

Horizontal Line Test

A

Every horizontal line intersects the graph of a one-to-one function at most once

40
Q

Existence of Inverse Functions

A

Let f be a one-to-one function on a domain D with a range R. Then f has a unique inverse f^-1 with domain R and range D such that

f^-1(f(x)) = x and f(f^-1(y)) = y

where x is in D and y is in R

41
Q

Finding an Inverse Function

A

Suppose f is one-to-one on an interval I. To find f^-1:

  1. Solve y = f(x) for x. If necessary, choose the function that corresponds to I.
  2. Interchange x and y and write y = f^-1(x)
42
Q

Logarithmic Function Base B

A

For any base b > 0, with b cannot equal 1, the logarithmic function base b, denoted y = logb x, is the inverse of the exponential function y = b^x

43
Q

Natural Logarithm Function

A

The inverse of the natural exponential function with base b = e is the natural logarithm function, denoted y = ln x

44
Q

Inverse Relations for Exponential and Logarithmic Functions

A

For any base b > 0, with b cannot equal 1, the following inverse relations hold:

  1. b^logbx = x (for all x > 0)
  2. logbb^x = x (for all real values of x)
45
Q

Properties of Logarithmic Functions

A
  1. Because the range of b^x is y: 0 < y < infinity, the domain of logb x is x: 0 < x < infinity.
  2. The domain of b^x is all real numbers, which implies that the range of logb x is all real numbers
  3. Because b^0 = 1, it follows that logb 1 = 0
  4. If b > 1, then logb x is an increasing function
46
Q

Change-of-Base Rules

A

Let b be a positive real number with b cannot equal 1. Then

b^x = e^xlnb, for all x and logbx = ln x/ln b, for x > 0

More generally, if c is a positive real number with c cannot equal 1, then

b^x = c^xlogcb, for x and logbx = logcx/logcb, for x > 0

47
Q

Radians (Rad)

A

The length of the arc associated with theta, denoted s, divided by the radius of the circle r

48
Q

Standard Position

A

An angle is in standard position if its initial side is on the positive x-axis and its terminal side on the line segment OP between the origin and P.

49
Q

Trigonometric Functions

A

Let P(x, y) be a point on a circle of radius r associated with the angle theta. Then

sin theta = y/r

csc theta = r/y

cos theta = x/r

sec theta = r/x

tan theta = y/x

cot theta = x/y

50
Q

Identities

A

Trigonometric functions that have a variety of properties that are true for all angles in the domain

51
Q

Period of Trigonometric Functions

A
  • The functions sin theta, cos theta, sec theta, and csc theta have a period of 2pi for all theta in the domain
  • The functions tan theta and cot theta have a period of pi for all theta in the domain
52
Q

Amplitude

A

A vertical stretch of abs value of A

53
Q

Period

A

A period of 2pi/abs value of B

54
Q

Phase Shift

A

A horizontal shift of C

55
Q

Vertical Shift

A

A vertical shift of D

56
Q

Inverse Sine/Arcsine

A

y = sin^-1x is the value of y such that x = sin y, where -pi/2 < or equal to y < or equal to pi/2 with a domain of x: -1 < or equal to x < or equal to 1.

57
Q

Inverse Cosine/Arccosine

A

y = cos-1 x is the value of y such that x = cos y, where 0 < or equal to y < or equal to pi with a domain of x: -1 < or equal to x < or equal to 1.

58
Q

Tan and Cotangent Function

A

y = tan-1 x is the value of y such that x = tan y, where -pi/2 < y < pi/2 with a domain of x: - infinity < x < infinity

y = cot-1 x is the value of y such that x = cot y, where 0 < y < pi with a domain of x: - infinity < x < infinity.

59
Q

Secant and Cosecant Function

A

y = sec -1 x is the value of y such that x = sec y, where 0 < or equal to y < or equal to pi, with y cannot equal pi/2 with a domain of x: abs value of x > or equal to 1

y = csc -1 x is the value of y such that x = csc y, where -pi/2 < or equal to y < or equal to pi/2, where y cannot equal 0 with a domain of x: abs value of x > or equal to 1.

60
Q

Periodic Functions

A

Functions whose value repeat every interval of some fixed length

61
Q

Period

A

The smallest positive real number that has the property of periodic functions

Calculus (2 decks)

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