Functions

Question 1

Domain

Answer 1

Set of x-values that yeild an output

Question 2

Range

Answer 2

Set of possible outputs of a function

Question 3

Independent variable

Answer 3

X-value, associated with the domain

Question 4

Dependent variable

Answer 4

Y-value, associated with range

‘Depends’ on x-value

Question 5

Graph

Answer 5

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

Question 6

Argument

Answer 6

The function expression

Represented by ‘f(x)’

Question 7

Vertical line test

Answer 7

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

Question 8

Interval notation

Answer 8

Exclusive ()

Inclusive [ ]

Question 9

Composite function

Answer 9

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

Question 10

Symmetric to x-axis

Answer 10

The graph is the same when flipped upside down, folded on x

Cannot be a function, fails the vertical line test

Question 11

Symmetric to the y-axis

Answer 11

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

Occupies adjacent quadrants

Question 12

Symmetric to the origin

Answer 12

Looks the same when rotated 180 degrees on the paper

Occupies diagonal quadrants

Question 13

Even functions

Answer 13

f(x)=f(-x)

Looks the same which viewed backwards

Question 14

Odd functions

Answer 14

f(-x)=-f(x)

Looks the same which viewed from the origin

Question 15

Polynomials

Answer 15

Algebraic functions represented by terms with descending powers

Question 16

Rational functions

Answer 16

Algebraic function in which one polynomial divided by another

Question 17

Algebraic Functions

Answer 17

Use only +,-, x, /, ^, or √

Question 18

Exponential functions

Answer 18

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

Question 19

Logarithmic functions

Answer 19

Transcendental functions in the form Log-base exponent

Question 20

Trigonometric function

Answer 20

Transcendental functions Involving trigonometric expressions

Question 21

Transcendental Functions

Answer 21

Non-algebraic functions

Question 22

Linear function

Answer 22

Algebraic function that take the form ‘y=mx+b’

Question 23

Peicewise functions

Answer 23

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

Question 24

Power function

Answer 24

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

Question 25

Root functions

Answer 25

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

Question 26

Function transformation

Answer 26

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

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

Question 27

Vertical stretch

Answer 27

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

Question 28

Vertical shift

Answer 28

Factor added or subtracted from function output

f(x)±d

Question 29

Natural exponential function

Answer 29

f(x)=e^x

e is the base in the exponential function

Question 30

Inverse function

Answer 30

The argument for f^(-1)(x)

Calculated by isolating the x-variable on the =

Question 31

One-to-one function

Answer 31

Each output has only one x-value

Use a ‘horizontal-line test’

Question 32

Horizontal-line test

Answer 32

Test to determine whether function is one-to-one

Question 33

Change of base formula

Answer 33

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

Question 34

Radians

Answer 34

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

Question 35

Angle measure, from radians

Answer 35

θ=s/r

s- radians
r- radius

Question 36

Hypotenuse

Answer 36

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

Question 37

Cosine θ

Answer 37

Cosθ= adj/hyp= x/r

Question 38

Sine θ

Answer 38

Sinθ= opp/hyp= y/r

Question 39

Tangent θ

Answer 39

Tanθ= opp/adj= y/x

Question 40

Cotangent θ

Answer 40

Cotθ= adj/opp= x/y

Question 41

Secant θ

Answer 41

Secθ= hyp/adj= r/x

Question 42

Cosecant θ

Answer 42

Cscθ= hyp/opp= r/y

Question 43

Reciprocal identities (tangent)

Answer 43

Tanθ= sinθ/cosθ

Question 44

Reciprocal identities (Cotangent)

Answer 44

Cotθ= cosθ/sinθ

Question 45

Reciprocal identities (Cosecant)

Answer 45

Cscθ= 1/sinθ

Question 46

Reciprocal identities (Secant)

Answer 46

Secθ= 1/cosθ

Question 47

Reciprocal identities (sine)

Answer 47

Sinθ= 1/cscθ

Question 48

Reciprocal identities (cosine)

Answer 48

Cosθ= 1/secθ

Question 49

Pythagorean Identities [Sin^2(θ)]

Answer 49

Sin^2(θ)=1-cos^2(θ)

Question 50

Pythagorean Identities [cos^2(θ)]

Answer 50

cos^2(θ)=1-Sin^2(θ)

Question 51

Pythagorean Identities [tan^2(θ)]

Answer 51

tan^2(θ)=Sec^2(θ)-1

Question 52

Pythagorean Identities [cot^2(θ)]

Answer 52

cot^2(θ)=csc^2(θ)-1

Question 53

Pythagorean Identities [csc^2(θ)]

Answer 53

csc^2(θ)=cot^2(θ)+1

Question 54

Pythagorean Identities [sec^2(θ)]

Answer 54

Sec^2(θ)=1+tan^2(θ)

Question 55

Double-half Angle formulas [sin^2(θ)]

Answer 55

sin^2(θ)=(1-cos(2*θ))/2

Question 56

Double-half Angle formulas [cos(2*θ)]

Answer 56

cos(2*θ)=cos^2(θ)-sin^2(θ)

Question 57

Horizontal Stretch

Answer 57

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

f(ax)

Question 58

Arc length (radians)

Answer 58

S=θ*radius

Question 59

Radius, from radians

Answer 59

Radius=Radians/θ

Question 60

Period (sec/cyc)

Answer 60

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

Question 61

Frequency (cyc/sec)

Answer 61

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

Question 62

Double-half Angle formulas [cos^2(θ)]

Answer 62

cos^2(θ)=(1+cos(2*θ))/2

Question 63

Inverse Trig functions

Answer 63

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

Question 64

Double-half Angle formulas [sin(2*θ)]

Answer 64

sin(2θ)=2sinθ*cosθ

Question 65

Horizontal shift

Answer 65

Factor added or subtracted from variable

f(x±b)

Question 66

Modeling Growth

Answer 66

Always as:
A(t)=Pe^(rt)
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

Question 67

Graphing complex trig functions

Answer 67

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

Question 68

Sinθ times the cosθ

Answer 68

Sinθ*cosθ=(1/2)sin(2θ)

Question 69

Reduction of cos^2(θ)-1

Answer 69

cos^2(θ)-1=1/2*cos(2a)

Question 70

Reduction of 1-sin^2(θ)

Answer 70

1-sin^2(θ)=1/2*cos(2a)

Question 71

Reduction of cos^2(θ)-sin^2(θ)

Answer 71

cos^2(θ)-sin^2(θ)=cos(2a)

Question 72

Reduction of 3sin(θ)-4sin^3(θ)

Answer 72

3sin(θ)-4sin^3(θ)=sin(3*θ)

Question 73

Reduction of 4cos^3(θ)-3cos(θ)

Answer 73

4cos^3(θ)-3cos(θ)=cos(3*θ)

Question 74

Function

Answer 74

An continuous curve for which every input has an output