Functions Flashcards Preview

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Flashcards in Functions Deck (74):
1

Domain

Set of x-values that yeild an output

2

Range

Set of possible outputs of a function

3

Independent variable

X-value, associated with the domain

4

Dependent variable

Y-value, associated with range
'Depends' on x-value

5

Graph

Set of all points (x,y) represented on a plane

6

Argument

The function expression
Represented by 'f(x)'

7

Vertical line test

When examining a graph, if there is more than one output for any one input, the curve cannot represent a function

8

Interval notation

Exclusive ()
Inclusive [ ]

9

Composite function

Function whose input depends on the output of a second function g(x)
Writen as (f o g)(x)
Or f(g(x))

10

Symmetric to x-axis

The graph is the same when flipped upside down, folded on x
Cannot be a function, fails the vertical line test

11

Symmetric to the y-axis

The graph looks the same if viewed backwards, folded on y
Occupies adjacent quadrants

12

Symmetric to the origin

Looks the same when rotated 180 degrees on the paper
Occupies diagonal quadrants

13

Even functions

f(x)=f(-x)
Looks the same which viewed backwards

14

Odd functions

f(-x)=-f(x)
Looks the same which viewed from the origin

15

Polynomials

Algebraic functions represented by terms with descending powers

16

Rational functions

Algebraic function in which one polynomial divided by another

17

Algebraic Functions

Use only +,-, x, /, ^, or √

18

Exponential functions

Transcendental functions in which the variable is an exponent to a given base.
Infinite domain
Range>0
As x→0, f(x)=1

19

Logarithmic functions

Transcendental functions in the form Log-base exponent

20

Trigonometric function

Transcendental functions Involving trigonometric expressions

21

Transcendental Functions

Non-algebraic functions

22

Linear function

Algebraic function that take the form 'y=mx+b'

23

Peicewise functions

A function in which the argument is different on a variety of intervals
Writen as f(x)={argument

24

Power function

Algebraic function in which the variable is raised to a given power

25

Root functions

Algebraic function in which the variable is down to a √ or ^(1/n)

26

Function transformation

y=cf(a(x-b))+d

a- horizontal stretch
b- horizontal shift
c- vertical stretch
d- vertical shift

27

Vertical stretch

Factor multiplied by the function output, (could be a fraction)
c(f(x))

28

Vertical shift

Factor added or subtracted from function output
f(x)±d

29

Natural exponential function

f(x)=e^x
e is the base in the exponential function

30

Inverse function

The argument for f^(-1)(x)
Calculated by isolating the x-variable on the =

31

One-to-one function

Each output has only one x-value
Use a 'horizontal-line test'

32

Horizontal-line test

Test to determine whether function is one-to-one

33

Change of base formula

Log-f(x) = [log-i(x)]/[log-i(f)]

34

Radians

Number of 'radius lengths' an arc completes
π for one full circle
Number of circles is described in trig-functions

35

Angle measure, from radians

θ=s/r

s- radians
r- radius

36

Hypotenuse

Longest side of the triangle
Radius when represented by a circle
H=√(x^2+y^2)

37

Cosine θ

Cosθ= adj/hyp= x/r

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Sine θ

Sinθ= opp/hyp= y/r

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Tangent θ

Tanθ= opp/adj= y/x

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Cotangent θ

Cotθ= adj/opp= x/y

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Secant θ

Secθ= hyp/adj= r/x

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Cosecant θ

Cscθ= hyp/opp= r/y

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Reciprocal identities (tangent)

Tanθ= sinθ/cosθ

44

Reciprocal identities (Cotangent)

Cotθ= cosθ/sinθ

45

Reciprocal identities (Cosecant)

Cscθ= 1/sinθ

46

Reciprocal identities (Secant)

Secθ= 1/cosθ

47

Reciprocal identities (sine)

Sinθ= 1/cscθ

48

Reciprocal identities (cosine)

Cosθ= 1/secθ

49

Pythagorean Identities [Sin^2(θ)]

Sin^2(θ)=1-cos^2(θ)

50

Pythagorean Identities [cos^2(θ)]

cos^2(θ)=1-Sin^2(θ)

51

Pythagorean Identities [tan^2(θ)]

tan^2(θ)=Sec^2(θ)-1

52

Pythagorean Identities [cot^2(θ)]

cot^2(θ)=csc^2(θ)-1

53

Pythagorean Identities [csc^2(θ)]

csc^2(θ)=cot^2(θ)+1

54

Pythagorean Identities [sec^2(θ)]

Sec^2(θ)=1+tan^2(θ)

55

Double-half Angle formulas [sin^2(θ)]

sin^2(θ)=(1-cos(2*θ))/2

56

Double-half Angle formulas [cos(2*θ)]

cos(2*θ)=cos^2(θ)-sin^2(θ)

57

Horizontal Stretch

Factor multiplied by the x-variable, (could be a fraction)
f(ax)

58

Arc length (radians)

S=θ*radius

59

Radius, from radians

Radius=Radians/θ

60

Period (sec/cyc)

Length of a single trigonometric cycle
Period=2π/B
Where B is y=sin(B*x)
Also Period=1/frequency

61

Frequency (cyc/sec)

Number of cycles that occurs per x-unit
Frequency=B/2π
Where B is y=sin(B*x)
Also Frequency=1/period

62

Double-half Angle formulas [cos^2(θ)]

cos^2(θ)=(1+cos(2*θ))/2

63

Inverse Trig functions

y=trig^-1(x)
x=trig(y)
Reflexive over the y=x line, to their original function

64

Double-half Angle formulas [sin(2*θ)]

sin(2*θ)=2*sinθ*cosθ

65

Horizontal shift

Factor added or subtracted from variable
f(x±b)

66

Modeling Growth

Always as:
A(t)=P*e^(r*t)
A- actual amount as a function of 't'
P- principal, the value with which you started
r- rate, new output units per unit of time
t- time

67

Graphing complex trig functions

1) Create graph with respect to time
2) Start at x=hShift. If sine, y=vShift. If cosine, y=amplitude+vShift
3) Calculate period from the frequency. Mark the above y-value at every frequency multiple on x
4) If sine, mark that y-value between the frequency multiples too. If cosine, mark those frequency midpoints with yValue-(2*amplitude)
5) If sine, mark the first frequency quarter point with (amplitude+vShift), alternating between positive and negative for each half-frequency measure thereafter. If cosine, mark the frequency quarter points with the y-value in the middle of the y-values on either side.
6) Connect all of the points with a smooth curve

68

Sinθ times the cosθ

Sinθ*cosθ=(1/2)sin(2θ)

69

Reduction of cos^2(θ)-1

cos^2(θ)-1=1/2*cos(2a)

70

Reduction of 1-sin^2(θ)

1-sin^2(θ)=1/2*cos(2a)

71

Reduction of cos^2(θ)-sin^2(θ)

cos^2(θ)-sin^2(θ)=cos(2a)

72

Reduction of 3sin(θ)-4sin^3(θ)

3sin(θ)-4sin^3(θ)=sin(3*θ)

73

Reduction of 4cos^3(θ)-3cos(θ)

4cos^3(θ)-3cos(θ)=cos(3*θ)

74

Function

An continuous curve for which every input has an output