Functions Flashcards
Domain
Set of x-values that yeild an output
Range
Set of possible outputs of a function
Independent variable
X-value, associated with the domain
Dependent variable
Y-value, associated with range
‘Depends’ on x-value
Graph
Set of all points (x,y) represented on a plane
Argument
The function expression
Represented by ‘f(x)’
Vertical line test
When examining a graph, if there is more than one output for any one input, the curve cannot represent a function
Interval notation
Exclusive ()
Inclusive [ ]
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))
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
Symmetric to the y-axis
The graph looks the same if viewed backwards, folded on y
Occupies adjacent quadrants
Symmetric to the origin
Looks the same when rotated 180 degrees on the paper
Occupies diagonal quadrants
Even functions
f(x)=f(-x)
Looks the same which viewed backwards
Odd functions
f(-x)=-f(x)
Looks the same which viewed from the origin
Polynomials
Algebraic functions represented by terms with descending powers
Rational functions
Algebraic function in which one polynomial divided by another
Algebraic Functions
Use only +,-, x, /, ^, or √
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
Logarithmic functions
Transcendental functions in the form Log-base exponent
Trigonometric function
Transcendental functions Involving trigonometric expressions
Transcendental Functions
Non-algebraic functions
Linear function
Algebraic function that take the form ‘y=mx+b’
Peicewise functions
A function in which the argument is different on a variety of intervals
Writen as f(x)={argument
Power function
Algebraic function in which the variable is raised to a given power
Root functions
Algebraic function in which the variable is down to a √ or ^(1/n)
Function transformation
y=cf(a(x-b))+d
a- horizontal stretch
b- horizontal shift
c- vertical stretch
d- vertical shift
Vertical stretch
Factor multiplied by the function output, (could be a fraction)
c(f(x))
Vertical shift
Factor added or subtracted from function output
f(x)±d
Natural exponential function
f(x)=e^x
e is the base in the exponential function
Inverse function
The argument for f^(-1)(x)
Calculated by isolating the x-variable on the =
One-to-one function
Each output has only one x-value
Use a ‘horizontal-line test’
Horizontal-line test
Test to determine whether function is one-to-one
Change of base formula
Log-f(x) = [log-i(x)]/[log-i(f)]
Radians
Number of ‘radius lengths’ an arc completes
π for one full circle
Number of circles is described in trig-functions
Angle measure, from radians
θ=s/r
s- radians
r- radius
Hypotenuse
Longest side of the triangle
Radius when represented by a circle
H=√(x^2+y^2)
Cosine θ
Cosθ= adj/hyp= x/r
Sine θ
Sinθ= opp/hyp= y/r
Tangent θ
Tanθ= opp/adj= y/x
Cotangent θ
Cotθ= adj/opp= x/y
Secant θ
Secθ= hyp/adj= r/x
Cosecant θ
Cscθ= hyp/opp= r/y
Reciprocal identities (tangent)
Tanθ= sinθ/cosθ
Reciprocal identities (Cotangent)
Cotθ= cosθ/sinθ
Reciprocal identities (Cosecant)
Cscθ= 1/sinθ
Reciprocal identities (Secant)
Secθ= 1/cosθ
Reciprocal identities (sine)
Sinθ= 1/cscθ
Reciprocal identities (cosine)
Cosθ= 1/secθ
Pythagorean Identities [Sin^2(θ)]
Sin^2(θ)=1-cos^2(θ)
Pythagorean Identities [cos^2(θ)]
cos^2(θ)=1-Sin^2(θ)
Pythagorean Identities [tan^2(θ)]
tan^2(θ)=Sec^2(θ)-1
Pythagorean Identities [cot^2(θ)]
cot^2(θ)=csc^2(θ)-1
Pythagorean Identities [csc^2(θ)]
csc^2(θ)=cot^2(θ)+1
Pythagorean Identities [sec^2(θ)]
Sec^2(θ)=1+tan^2(θ)
Double-half Angle formulas [sin^2(θ)]
sin^2(θ)=(1-cos(2*θ))/2
Double-half Angle formulas [cos(2*θ)]
cos(2*θ)=cos^2(θ)-sin^2(θ)
Horizontal Stretch
Factor multiplied by the x-variable, (could be a fraction)
f(ax)
Arc length (radians)
S=θ*radius
Radius, from radians
Radius=Radians/θ
Period (sec/cyc)
Length of a single trigonometric cycle
Period=2π/B
Where B is y=sin(B*x)
Also Period=1/frequency
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
Double-half Angle formulas [cos^2(θ)]
cos^2(θ)=(1+cos(2*θ))/2
Inverse Trig functions
y=trig^-1(x)
x=trig(y)
Reflexive over the y=x line, to their original function
Double-half Angle formulas [sin(2*θ)]
sin(2θ)=2sinθ*cosθ
Horizontal shift
Factor added or subtracted from variable
f(x±b)
Modeling Growth
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
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
Sinθ times the cosθ
Sinθ*cosθ=(1/2)sin(2θ)
Reduction of cos^2(θ)-1
cos^2(θ)-1=1/2*cos(2a)
Reduction of 1-sin^2(θ)
1-sin^2(θ)=1/2*cos(2a)
Reduction of cos^2(θ)-sin^2(θ)
cos^2(θ)-sin^2(θ)=cos(2a)
Reduction of 3sin(θ)-4sin^3(θ)
3sin(θ)-4sin^3(θ)=sin(3*θ)
Reduction of 4cos^3(θ)-3cos(θ)
4cos^3(θ)-3cos(θ)=cos(3*θ)
Function
An continuous curve for which every input has an output
