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YGK Functions

YGK Classifications of Functions

An association between input values and output values, in which each input value is associated with exactly one output value. Function
Functions made of terms added together, in which each term is a number times a product of variables raised to nonnegative-integer powers. polynomial
For instance, 3x2y and –πx7y2z3 are each terms, so 3x2y – πx7y2z3 is a polynomial
Much of math is concerned with polynomials involving only one variable, such as –x3 + 2x2. The number at the beginning of each term is called a ___________ coefficient
Polynomials can be classified according to their number of terms: a polynomial with one term, like 2x or –12x2, is called a ______________ monomial
polynomial with two terms is called a binomial
polynomial with three terms is called a trinomial
Each polynomial has a ________. For polynomials of one variable, the ___________ is the largest exponent on the variable, so for the polynomial 4x3 – x2, the __________ is 3 degree
Since simple numbers (_______________) can also be written as the same number times any variable to the zeroth power (that is, 6 is the same as 6x0), numbers are also considered terms and polynomials constants
A polynomial with degree 1, like 3x, is called _________ linear
a polynomial with degree 2, like 3x2 – 8x + 4, is called __________________ quadratic
a polynomial with degree 3 is called ___________ cubic
4th degree polynomial quartic
5th degree polynomial quintic
6th degree polynomial sextic
the statement that every single-variable polynomial, other than constants, has a root in the complex numbers, which means that if f(x) is a polynomial, then the equation f(x) = 0 has at least one solution where x is some complex number. Fundamental Theorem of Algebra
The statement that there is no way to find a formula for the solutions of all quintic or higher-degree polynomials, if the formula must be based on the traditional operations (addition/subtraction, multiplication/division, and exponentiation/taking roots) The Abel-Ruffini theorem (Abel's impossibility theorem)
That impossibility is the topic that began an area of study called _____________ theory, which is part of abstract algebra. Galois [gal-wah]
The graph of a quadratic equation will be in the shape of a ______________ that opens straight up (if the coefficient on the x2 term is positive) or straight down (if that coefficient is negative) parabola
It is possible to find the roots of a quadratic through these 4 methods graphing it, factoring it, completing the square, or using the quadratic formula
If the quadratic is in the form ax2 + bx + c, then the expression b2 – 4ac, which appears in the quadratic formula, is called the ___________ discriminant
If the discriminant is positive, the quadratic will have ______ _______ roots two real
if the discriminant is zero, the quadratic will have ___ _______ root (said to have a multiplicity of 2) one real
if the discriminant is negative, the quadratic will have ____ _____ ____ __________ roots two non-real complex
if the coefficients of the quadratic are real numbers, the complex roots will be ________________ of each other conjugates
Function in which one polynomial divided by another polynomial. Examples therefore include 1/x, x2/(x – 3), and (x2 + 1)/(x2 – 1) rational functions
The denominator polynomial cannot be the zero polynomial, because dividing by zero is _____________ undefined
Every polynomial can be considered to be a ____________ function because 1 is a polynomial and dividing by 1 doesn’t change an expression rational
It is often instructive to study the __________________ of rational functions, which are places in which their graphs approach a line (or occasionally other shape), usually getting infinitely close to but not crossing it. asymptotes
Asymptote analysis may require performing long division on the numerator and denominator polynomials to find their ___________ _____________ ____________ greatest common factor
functions whose graph repeats a pattern (specifically, the graph has translational symmetry) Periodic functions
Technically speaking, a function of one variable f is periodic if f(x+p) = f(x) for every x in the domain of the function and some positive number p, which is called the _____________, because the graph repeats itself every p units period
While the trigonometric functions are the periodic functions most commonly encountered by high school math students, some other functions like __________ _____________ are also periodic; in general, functions representing waves tend to be periodic. triangle waves
a way to rewrite (almost) any periodic function in terms of only sine and cosine functions. Fourier series [fur-ee-ay]
functions that represent relations between angles and sides of triangles. trigonometric functions
Trigonometric functions are often illustrated using points and segments related to a circle of radius 1 centered at the origin, called the ______ ___________ unit circle
By far the most commonly discussed trigonometric functions are the sine, cosine, tangent, cosecant, cotangent, secant
The graphs of sine and cosine are ____________ of each other translations
the _____________ function equals the sine function divided by the cosine function tangent
the cosecant, secant, and cotangent functions are the _________________ of the sine, cosine, and tangent functions reciprocals
the inverse functions of the trigonometric functions. (Note that “inverse” in this case refers to a function that “undoes” another, not to “multiplicative inverse,” also called “reciprocal.”) inverse trigonometric functions
They are sometimes just called “inverse sine,” etc, and are also given with the prefix __ arc (arcsine, arccosine, arctangent, arccosecant, arcsecant, and arccotangent.)
functions that do not repeat any outputs. Injective functions, or injections
For instance, f(x) = 2x is _______________, because for every possible output value, there is only one input that will result in that output injective
On the other hand, f(x) = sin(x) is not ____________, because (for instance) the output 0 can be obtained from several different inputs (0, π, 2π, and so on) injective
If you have the graph of a function, you can determine whether the function is injective by applying the _________________ ____ _____; if no horizontal line would ever intersect the graph twice, the function is injective. horizontal line test
functions that achieve every possible output surjective functions or surjections
For instance, if you are thinking of functions whose domain and codomain are both the set of all real numbers, then f(x) = tan(x) is ____________ because every real number is an output for some input. surjective
A function that is both injective and surjective is called ____________ or ____________ bijective or bijections
If a function is bijective, then it has an _____________ Furthermore, a function can only have an ____________ if it is bijective. inverse
satisfy the rule f(–x) = f(x) for every x in the domain of the function. Even functions
The graph of an even function has _____________ symmetry over the y-axis reflection
named because if a polynomial’s exponents (on the variable) are all even, then the polynomial is an ______ ____________; for instance, x2, 3x6, and –x8 + 7x4 even functions
There are other even functions, though, such as the ______ and absolute value functions. cosine
There are other even functions, though, such as the cosine and ____________ __________ functions. absolute value
satisfy the rule f(–x) = –f(x) for every x in the domain of the function odd functions
The graph of an ______ function remains the same when it is rotated 180° around the origin odd
named because if a polynomial’s exponents (on the variable) are all odd, then the polynomial is an ______ __________; for instance, x3, 4x7, and –x5 + 2x3 odd function
There are other odd functions, such as the _______ and cube root functions. since
There are other odd functions, such as the sine and _____ _____ functions. cube root
Only one function is both even and odd the zero function, f(x) = 0.
functions whose of the form f(x) = bx, where b (called the base) is a positive number other than 1. exponential functions
Exponential functions are used to model unrestricted __________ (such as compound interest, and animal populations with unlimited food and no predators) growth
Exponential functions are also used to model _______ (such as radioactive decay) decay
The phrase “the exponential function” refers to the function f(x) = ex, where e is a specific irrational number called _______ __________ about equal to 2.718 Euler's number
Exponential functions have the interesting property that their __________________ are proportional to themselves. derivatives
functions of the form f(x) = logbx, where b is again a positive number other than 1 (and again called the base). They are the inverses of the exponential functions with the same bases Logarithmic functions, or logarithms
______________ functions are used to model sensory perception and some phenomena in probability and statistics Logarithmic
The phrase “the logarithm” can refer to a logarithmic function using the base ___ (especially in computer science; this is also called the binary logarithm) 2
The phrase “the logarithm” can refer to a logarithmic function using the base 2 (especially in computer science; this is also called the ________ logarithm) binary
The phrase common logarithm usually refers to the logarithm base ___ 10
The phrase natural logarithm refers to the logarithm base __ e
studied in calculus, these are functions where the limit approaching each point equals the function’s value at that point. In particular, there are no holes, jumps, or asymptotes “in the middle of the graph.” continuous functions
Created by: Mr_Morman