math 20 dec exam
product law
a^m x a^n = a^m+n
quotient law
a^m / a^n = a^m-n
power rule
a^m)^n = a^mn
distributivity over products
(ab)^m = a^m b^m
distributivity over quotients
(a/b)^m= a^m/b^m, b cant be zero.
power of zero
a^0 = 1, a cant be zero.
what is a radical
A radical n√x is a special symbol used to denote the nth root of some number x. ex: the square or second root of 4 is 2, because 2^2= 4.
a: (a n√x)
coeffiecient, when it has no coeffiecient, assume it is 1.
n: (a n√x)
index
x: (a n√x)
radicand
Multiplication property of a radical
n√ab = n√a x n√b
ex) √24= √4x6 = √4 x √6= 2√6
entire radical
An entire radical is an expression of the form a n√x where a must be 1 or -1. ex) 3√17 is an entire radical, but 4 2√2 is not.
mixed radical
A mixed radical is an expression of the form a n√x where a cannot be 1 or -1. ex) 5 2√9 is a mixed radical.
when is a radical in simplest terms
we say that a radical expression, a n√x is in simplified terms if the radicand does not contain any factors that are perfect powers of n.
rewrite a radical in exponential notation
let x∈R and n∈N, then n√x = x^1/n.
that is, we can rewrite a radical expression in exponential notation. ex) √5 = 5^1/2
index always goes on the bottom.
Negative exponents
if x∈R where x≠0 and n∈Q, then x^-n is defined as the reciprocal of x^n, that is x^-n = 1/x^n
you don’t’ take the reciprocal of the exponent, if it’s a radical.
negative is always in the denomnator, does not come along
cartesian product
The cartesian product of 2 sets A and B, denoted by A x B, is the set of ordered pairs of the form (a,b), where a∈A and b∈B. ex) AxB= {(x,1), (x,2), (y,1), (y,2)}
cartesian plane
The cartesian plane (aka coordinate plane) contains all possible pairings of the real numbers, if the cartesian plane represents RxR=R^2.
relation notation
A relation between two sets A and B is a subset of AxB denoted by R.
domain
The domain of a relation R is the set of all elements "a" such that (a,b)∈R. first component.
range
The range of R is the set of all elements "b" such that (a,b)∈R. second component.
what is a function
If R is a relation on the sets A and B, then it is a function if for all elements a∈A, there is exactly one element b∈B such that (a,b)∈R. so, the input can only be sent to one element in the output.
function notation
Function notation: f(x)
ex) C=P+2(P) becomes C(P)=P+2(P)
Hole in the graph
interval notation: (a,b) U (b,c)
inequality notation: a<x<b OR b<x<c
slopes of parallel lines
slopes of parallel lines always have the same slopes.
slopes of perpendicular lines
slopes of perpendicular lines are negative reciprocals of each other
slope intercept form
slope intercept form: y=mx+b, linear functions
m: slope, b: y intercept
slope of a line
The slope of a line is a number representing the steepness of the line.
Slope=
Slope= vertical change/horizontal change = rise/run. always reduce.
slope equation
slope= rise/run = y2-y1/x2-x1. order of x and y terms doesnt matter.
horizontal and vertical lines slope
horizontal lines always have a slope of zero because there is no vertical change. the equation is always y=#. Vertical Lines have an undefined slope because there is no horizontal change. The equation is always x=#.
steepness of slope
The greater the number the steeper the slope. The sign indicates the direction on the line not its steepness!
discrete set of numbers
discrete set of numbers is simply a set in which the values are separated or "disconnected” from one another, e.g. the set of integers is discrete
continuous set of numbers
a continuous set of numbers is “connected", e.g. the set of real numbers is continuous. This means that if we choose any two values in a continuous set, we can always find another value in between the initial two that we chose.
Continuous sets denotation
Continuous sets are denoted by intervals. For instance, the real numbers can be denoted by R or by (-∞, ∞).
closed interval
A closed interval, [a,b] contains all real numbers between a and b including a and b themselves.
In other words, [a,b] contains all real numbers x such that a ≤ x ≤ b.
open interval
An open interval, (a,b) contains all real numbers between a and b, but not a and b themselves.
In other words, (a,b) contains all real numbers x such that a < x < b.
interval order
interval order: smallest to biggest.
undefined functions
Some functions are undefined at certain values. For instance, consider the function f(x) = 1/x
Since we cannot divide by zero, f(x) = 1/x is undefined at zero. We call this a discontinuity at zero.
undefined graph
When a graph is undefined at a given value, say at value a, where a is some real number, we call. this a hole in the graph. A hole can be represented by an open circle.
density property
Density property. In simple terms, it states that given any two real numbers 'a' and 'b' with a < b, there exists a rational number 'r' such that a < r < b. ex is real numbers.
negative variables in power rule
odd variables during power rule stay negative, even ones do not because the coeffiecient is technically -1, not negative variable.
do you take reciprocal of coefficient when getting rid of negative exponents?
no
do negatives stay in the radical?
no