Functions

Independent Variable

Horizontal (x-value) - Can take any value

Dependent Variable

Vertical (y-value) - Depends on x value

Ways to represent independent/dependent

Mapping (choose an x, get the y)

Graphically

Function

Any independent value that produces one, and only one dependent value

Relation

Every graph, mapping, or equation is a relation. Independent values can produce one or more dependent values

Square Root Function vs. Square Root Operation

SR Function: y = \sqrt{x} (given, not produced, only positive)

SR Operation: y^2=x → y = \sqrt{x} (produced/inteoduced square root, +-, not function)

Vertical Line Test

Make an imaginary vertical line move across the cartesian plane. If it meets the curve more than once, it is not a function

Ordered Pair

Ex. {(3,6), (4,9), (5,2), (4,7), (2,6), (8,9)}

This set is a relation (x produce 1+ y)

Domain

The domain of a function or relation is the set of all independent values the function or relation can take

Number Systems (Continuous/Discrete)

Continuous: Real Numbers (Full line, every number touched) ~ Infinity numbers between every integer

Discrete: Integers (Dots, broken, only numbers listed are touched)

D: {x|xER}

R: {y|yER}

Domain is a set of x elements or real numbers such that x is an element of real numbers

Range is a set of y elements or real numbers such that y is an element of real numbers

* List values from least to greatest

End Point Possibilities

• - Is included

- - Is included

\omicron - Not included

Limit

When a line almost touches a point/number but doesn't (verryy close)

Range

The range of a function or relation is the set of all dependent values the function or relation can take

Represent Domain/Range Continuously/Discretely

Continuous: As a set

Discrete: As a list (Cuz its integers only)

f(x)

Function of X (not multiplication) = y

Parent Function

Non-transformed, non-translated functions

Parent Function - Linear

y = x

over 1, down 1 (As parent function)

Parent Function - Quadratic

y = x^2

over 1, down 1 (As parent function)

Parent Function - Absolute Value

y = |x|

Parent Function - Inverse

y = 1/x

Piecewise Function

One function with multiple pieces to describe it

Asymptote

A line which a curve never touches (Gets super close or far)

Parent Function - Square Root

y = \sqrt{x}

Parent Function - Exponential

y = b^{x}

b = base

asymptote at y = 0

Parent Function - Logarithmic

y = \log b^{x}

* b is subscript

Parent Function - Sine

y = \sin x

Parent Function - Cosine

y = \cos x

Parent Function - Tangent

\tan x=\sin x / \cos x

Sine/Cosine Rule

Always in between 1 and -1

How to find the Inverse of a Function?

Transpose the x and y values then resolve for y (Only one x and one y)

f^{-1} (x)

Transformations/Translations - a

a - vertical stretch/compress and inversion (Affect x-values)

If a > 1, it is a vertical stretch

If a is in between 0 and 1, it is a vertical compression

If a < 0, it is a vertical reflection (over x-axis)

* By a factor of

Transformations/Translations - c

c - Vertical Translation (Affect y-values)

If c > 0, it is vertically translated up by c

If c < 0, it is vertically translated down by c

Transformations/Translations - d

d - Horizontal Translation (Affect x-values)

If d > 0, it is a translation right by d

If d < 0 , it is a translation left by d

Transformations/Translations - k

k - Horizontal Stretch/Compression ~ cant do with simple curves (Affects x-values)

If k > 1, it is horizontally compressed by a factor of 1/k

If k is in between 0 and 1, it is a horizontal stretch by a factor of 1/k

If k < 0, it is a horizontal reflection (over y-axis)

* By a factor of

Will the inverse of a function always be a function?

For lines yes, anything with bumps (curve that changes direction) in it no.

Mapping Equations

y2 = a y1 + c

x2 = 1/k x1 + d

Rational Function

Looks like f(x) = g(x)/h(x)

Restriction

independent values on the domain of a function that it can not take because it would break mathematical rules (Ex. Division by 0 or square root of negatives) 

How to find restrictions

Factor higher order forms to linear form

How to factor higher-order polynomials?

Look for groups (usually of 2), use factoring algorithm

Factoring Algorithm

1) Common Factor

2) Difference of Squares (Make sure both numbers have square roots, open two brackets, put SR of each in both, one +, one -)

3) Perfect Square Trinomial (Open one bracket, square it, make sure last value is positive and both values on the ends have square roots that equal the middle # when multiplied and then doubled)

4) Decomposition (Break down middle value using its factors)

Produces hole vs. asymptote 

Hole: Restriction was not in final/reduced answer

Asymptote: Restriction remained in final answer/did not get reduced out