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

38

## Sine θ

### Sinθ= opp/hyp= y/r

39

## Tangent θ

### Tanθ= opp/adj= y/x

40

## Cotangent θ

### Cotθ= adj/opp= x/y

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## Secant θ

### Secθ= hyp/adj= r/x

42

## Cosecant θ

### Cscθ= hyp/opp= r/y

43

## 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