math 108
Percentage change formula
(change in price / initial price) * 100
relationship between (f*g)(x) and f(X)*g(X)
(f*g)(x)= f(X)*g(X)
multiplying functions
(f*g)(x)=f(x)*g(x)
inverse function
(f*g)(x)=x for all the x's in the domain of g and (g*f)(x)=x for all tx in the domain of f
the sum of functions
(f+g)(x)=f(x)+g(x)
subtracting functions
(f-g)(x) = f(x) - g(x)
dividing functions
(f/g)(x) =F(X)/G(X)
Limit Rules
- Lim f(x)=L when x approaches infinity means that f(x) gets close to l when the values of x that are positive enough - Lim f(x)=L when x approaches negative infinity, when x values are negative enough -Lim f(x)= infinity when x approaches a on its negative side f(x) is large when we choose values of x close but less than a limf(x)= negative infinity when x approaches a on its negative side , f(x)= can be small id we choose values of x that are close to a,but less than
wholes in functions
- find common factors in the numerator and denominator -see what x cannot equal in the denominator
finding the slope of secant line
- subtract the y value of the of the ending point - the y value of the center point divided by the x value of the endpoint - the x value of the center point -change in y/change in x
rationalizing
- used in fractional equations with square roots -multiple the equation by the square root and the negative of the number that is being added or subtracted to the square root part of the equation
locally linear
- when you zoom in on a point on a graph, the line that it is on become more and more straight (the tangent of a line at the point is linear in small intervals
how to see if a function has a vertical asymptote
-If the functions graph is separated with a vertical line
to solve negative integer exponents
-X=0, cannot be the domain of the function -factor out the lowest common factor of x(the biggest negative exponent) and solve
finding the tangent line centered around a point
-draw the tangent through the point on the graph
estimating instantaneous velocity
-find the average velocity at time intervals that include a point below and above the the time at which you want to find instantaneous velocity ex: at 2 ,[2,3] and [1,3] plug into derivative formula
to see if a function is negative or positive and how to write the postive and negative functions down?
-function is positive wherever the graph lies above the horizontal axis and is negative where ever the graph lies below the x axis -for positive functions a point that is above the x axis and then another point above the x axis right before the graph starts to go below the x axis - for negative function you take a negative point and see where the next point x=0 before crossing above the x axis -for both do this for all points
increasing and decreasing intervals
-increasing: if the output of f increases as the input increases -decreases: the output of f decreases as the input decreases
One-sided limits rules
-lim f(x)=l, then lim f(x)=l when x approaches a^- and when x (X->a ) approaches -if f(x)when x approcahes a on its negative side is not equal to f(x)when x approaches a on its positive side, then limit of f(x) when x aproaches a does not =L
slope-intercept form
-slope times x +plus the y intercept y=mx+b
how to make a repeating number a fraction?
-take the number and make sure the first set of repeating digit is on the left side of the decimal -make x+ 100x then subtract this equation from an equation with 10x =the repeating number with its repeating digits immediately to the right of the decimal
composite functions
-two limits go to the same equation -plug in both limits into the equation
obvious limits
-when x is in the denominator and dis n value the value gets smaller and smaller. - when is in the denominator and it is going towards - infinity, x gets smaller and smaller - if x is and is raised to the positive n then is goes to infintiy on both sides - if x is raised to an odd positive integer then and x goes towards -00, then x goes to negative infinity and vice-versa
horizontal asymptote rules
1. bottom degree is bigger than top degree, then y=0 5x^4/3x^5 2. top degree is bigger than bottom degree, it goes to -oo or infintiy 5x^5/3x^3 3. if they are the same in the denominator and numerator then it equal the leading coefficient 3x/4x=3/4
limits with fractional equations
1. factor the top so the denominator does not equal zero
finding the tangent line of an equation at a point
1. find the slope of the tangent by finding the slope of two points surrounding the point 2. plugging in x and f(x) into the tangent line equation
word problem into piece wise linear fuction
1. for the first function plug in the points with point slope form 2. if the function changes at one point then plug in the new points with 2 point slope form
how to solve a quadratic equation?
1. get one side to equal 0 2. factor 3. solve for x's
piece-wise linear function on a graph
1. look at where the x value applies for each linear function 2. plug in those values into the linear function
point of intersection of two lines
1. make them equal to each other 2. solve the equation
Exponent Rules
1. multiplying the same expression with exponents add the exponents together Ex: x2x3 = x5 2. number times another number in parenthesizes to a power both of the number individually to that power Ex: (xy)3 =x3y3 3. (xm)n = xmn 4. xm/xn = xm-n 5. xy/2 = The squared Root of xy
limit with table of values
1. plug in the inputs into calculator 2. plug in list into the formula 3. get outputs
Evaluating Limits
1. plugin what the limit into the equation to find the y value of the limit limit x^2=a^2 x->a
zeros of the compostion function graph
1. see the input function equals 2. and then see what the y values of the output function
domains of a table function divided by another table function
1. see what x values the tables share 2. see if the outputs of those x values are divisible
graph to piece wise linear function
1. see where the intervals are 2. find the slope 3. plug in original point into two point slope form
finding the zeros of a composition of functions
1. see where the x intercepts on the first graph 2. plug in the x intercepts and put them on the y axis of the 2nd function and draw a horizontal line at those points 3.then see which points cross the horizontal lines
algebra to find limits
1. take highest power from denominator 2. factor it out of the denominator and numerator 3. see what each part of the function goes to to find the limit
4 ways to represent a function
1. verbally:sentence form of input and output 2. nuremically by a table: gives inputs and outputs 3. Graphically: points on the coordinate 4. analytically: equation form
to find where the tangent line is horizontal
1.calculate the vertex =-b/2a
how to solve a derivative with a function that is a fraction
1.cross multiple the top fraction and combine them together 2. multiple the h on the bottom fraction to the denominator of the top fraction 3. take out the h from the equation
way to solve quadratic equations?
1.factoring: 2.extracting roots 3. completing the square: take half of b and square it added it and subtract it to the equation (x2 + 4x + 4 = -1 + 4)
finding the x's in a composition of functions
1.look for what values equal the output functions sum 2. see which what values of the input function make it true
rules for finding the "implied" domain of functions
1.make sure denominator does not equal ex: 2x/x, x cannot equal 0 2.if the square root in a quadratic equation there are no solutions
Reciprocal Function
1/x
set
A collection of elements
vertical line test
A test used to determine whether a relation is a function by checking if a vertical line touches 2 or more points on the graph of a relation
rational numbers
All positive and negative integers, fractions and decimal numbers.
Intermediate Value Theorem
IF a function is continous and interval [a,b] is included and c is a point between a and b
whole numbers
Natural numbers ( counting numbers) and zero; 0, 1, 2, 3...
irrational numbers
Numbers that cannot be expressed as a ratio of two integers. Their decimal expansions are nonending and nonrepeating.
Compostion of Functions (algebraically)
One function becomes the domain of the other ex: (f*g)(x)= f(g(X))
Intergers
Whole numbers and their opposites (. . . -3, -2, -1, 0, 1, 2, 3. . .)
vertical line slope
X=a no slope/ undefined slope
function
a function is a input(independent) output(dependent) relationship, where the input only has one output
piece-wise linear function
a function is written using two or more linear expressions - cannot have two linear functions that meet at one point have the same brackets or inequalityEX: x greater than or equal to 5 and x is greater than 5 but less than or equal to 10 -
defined function
a function that is locally linear and non vertical
discontinuous function
a functions who's graph has a whole, gap, or vertical asymptote - (from inffintiy to the xvalue of (gap, whole, vertical asymptote)
tangent
a line that hits the graph once - underestimate of the slope
range
all acceptable outputs of a function. R(0,1,2,3,4)
Explicit list Notation
all the number within a set ex: A=(1,2,3)
rules of sets
any elements with in another bracket in the set count as one element an empty bracket with another empty bracket in it counts as 2 elements in the set
Derivative of a function
as change in x heads towards 0 from both side = the slope
writing the vertical asymptote Limits
as x -> asymptote on positive side , the function->infinity or negative infinity as x -> asymptote on the negative side, the function->infinity or negative infinity
average velocity formula
change in distance/change in time
average rate of change of a function
change in y/ change in x -ex: avg rate of change of f on [a,b]= ( f(B)-f(a))/ b-a
ends of a graph derivative
derivative does not exist because they are not continuous on both sides
derivitive of corners/cusps
do not exist
shifting a graph up or down
down: f(x)=x^2 -2 up: f(x)=x^2 +2
Natural Numbers
ever positive counting number above 0 (1,2,3,4,5,6)
General Form
everything is on one side ax+By+c=0
composition of functions
f ( g(x)) inputting g function into the f function or f*g
instantaneous rate of change
f'(a)=f(a+h)-f(a)/h
average rate of change
f(b)-f(a)/b-a
identity function graph
f(x)=x
rules of puting fuctional noation into sentences?
f(x*#)=input time the number 2f(x)= output times two f(x)+1= output +1
average velocity formula
f(x2)-f(x1)/x2-x1
change in price
final price - initial price
symmetric about the origin
flipped around the y axis and flipped around the x axis - ex:cubed functions, receprical function, and identitive functions
even functions
functions that are graph is symmetric around the y axis f(x)=f(-x) absolute value and reciprical graphs
inverse functions
functions that are the opposite of each other ex: (g*f)(x) and (f*g)(x)
conditional equation
has multiple answers ex:x^2+4
continuous function
has no gap, whole, or vertical asymptote - x to x values where there are no gaps,
Horizontal Asymptote`
horizontal line on the y axis in the middle of a graphed function
Removable Discontinuities
if a function WITH A HOLE can be made continuous by putting a whole
to find where the function is positive at x
if the output value f(a) is positive
to find if the function is negative at x
if the output value is negative
identity function
inputs equal the outputs ex: f(x)=x
interval
is any set of real numbers with one exception that any two numbers in the set has to include the number in between them
Distance Formula
is from the Pythagorean theorm d = √[( x₁ - x₂)² + (y₁- y₂)²]
undefined function
is not continuous, has a cusp, or id locally linear but is vertical
f prime of a number
is the slope of a tangent line going through a point -f'(x)
y intercept of a function
is where does y= when x is o
secant line
line of slope/ difference equation - f(x2)-f(x1)/x2-x1 -overestimate of the slope
Differentiability
locally linear
finding where if the graph is going to be negative or positive when it approaches the asymptote
looking at the part of the graph that is closet the the asymptote
slope formula
m=y1-y2/x1-x2
how to solve a quadratic equation by factoring?
make one side 0 then factor
finding the point on the graph when you have the tangents slope
make the derivative is equal to the slope
finding the derivative of a function that has a square root in its numerator.
multiple the top and bottom by the numerator but the do the the opposite of the what the -f(x) is in the equation ex:
linear equation
no variable can be hdivided or squared x+b=c
what are function and what are not functions?
not a function: y^2+x^2=16 each input has two outputs -when the absolute value is around y -
corner/cusp
not locally linear at point on graph - ex: absolute value function
rational number
numbers that have a repeating block of digits or terminate
Description Notation formula
object : (or I) singular description ex: x: 2x=7
theorum(sandwich or squeeze )
one function is in-between two others and shares a point
split function
outputs are different for different intervals of the domain f(x)= 2x-1, x is less than 2 x squared -2, x greater than or equal to 2 you would only plug in # into the the first equation if x is less than 2, but more if x is greater than or equal to two.
vertex of a vertex form equation
plug in 0 for x and solve ex: y=2(x-6)^2- 4/5 vertex:(6,4/5)
seeing if a point on a function exist using the derative
plug in x value into derivative formula and solve
how to solve a quadratic equation with the quadratics formula?
plug the equation into the formulA
polynominal =0
polynomials that p(x)=0 x is a critical or key number
when an input equals the output
q(x)=x
positive slope
rises from left to right
negative slope
rises from right to left m=-2 : up two, left 1 m= -1/2 up one, left two
interval notation set notation and picture/sketch notation
set notation: an expression that uses inequalities to describe subsets of real numbers ex: (x:X greater than or equal to three) interval notation: Limits of number in a set ex: (7,20) sketch :
shifting the graph left or right
shift left : y= (x+2)+z shift right: y=(x-2) +z
odd functions
symmetric around the origin f(-x)= -f(x)
finding the x intercepts of the product of 2 quotients
take the numerator and make it =0, ex:(2x)(x-3) x=0, x=3
translating points in a plane
take your original points and add the translation to your point ex: move 5 left and 2 down from point (6,4)= (11,2)
Horizontal Test math
tells us if the function is a one to one function by seeing if two or more points have the same y value
when a interval is increasing faster than the other interval s
the interval is at a more rapid rate than the other intervals
What does epsilon stand for?
the number is included in the set
a graph is vertically stretched or horizontally compressed
the original function is being multiplied by a number bigger than 0 y=2x^2 horizontally compressed by a half
domain
the set of all acceptable inputs of a function. ex: D(012345)
real numbers
the set of rational and irrational numbers
parallel lines
the slope of two lines are the same
perpendicular lines
the slope two lines are negative reciprocals of each other
when f(x) is differential
then it is continuous
midpoint formula
to find the mid point of 2 points on a graph (x₁+x₂)/2, (y₁+y₂)/2
sentences into functional notation
triple the input and subtract 4 from the total f(x)=3x-4
Union and Intersection of Sets
union: in between two sets that say everything within the 2 sets are included in the number line intersection:what number are included in both sets e
limit principle
what you need to do to the limits , just plug in the yvalues of the limits into the equation
Non-Removable Discontinuities
when a function has a asymptote or gap
quadratic formula rules
when the equals 0 it has 1 solution when it greater than 0 then it has 2 solutions when it the equation equals less than 0 then there are no solutions
graph is vertically compressed
when the original function is being multiplied by a number greater than 0 and less than 1. y=1/2x
reflection across the y axis
when the points are equal distance from the y axis
vertical asymptote
when the x or the denominator equals 0
graphs are neither even or odd functions when
when there are more than 1 x intercept -the inputs and the their recipricals should have the same output or the reciprical outputs.
two-point form
when there are two points on a line y-y1 = (y2-y1 / x2-x1) (x-x1)
the domains of two function tables
where the inputs (x's are the same)
constant function
where the numbers in the domain all go to the same output
reflecting over the y and x axis
x axis reflection: (x,y)= (x,-y) y axis reflection: (x,y)= (-x,y)
vertex of standard equations
x= b /-2a THEN PLUG IN X IN THE EQUATION
Squaring function
x^2
cubed function
x^3
Initial Value Formula
y value of point+slope (x- the intial xvalue of the point Y=yo+ m(x-x0)
derivative tangent equation
y-y1= m(x-x1)
tangent line equation
y-y1=m(x-x1) - plug in the center point in and slope of tangent
point slope formula
y-y1=m(x-x1) where m is the slope and (x1,y1) is a point on the line
writing horizontal asymptotes
y= asymptote as x +or - ->approaches 00 or -00,f(x)->___
Quadratic Standard Form
y= ax^2+bx+c
factored form of a quadratic equation
y=a(x-d)(x-e) THE VERTEX IS x+x/2a
Quadratic Vertex Form
y=a(x-h)^2+k -If h is negative the graph moves right and if h is postive the graph moves left
horizontal line
y=b where b is the y-intercept of the line
How to solve a quadraticequation by completing the square?
you factor the expression take the square root of both sides and solve
to solve positive integer exponents
you have to factor out the lowest common factors and solve
how to graph a number line with a polynomial?
you have to plug in values that are inbetween , lower and higher tpo the key or critical values . this is called the domain
the domain of a combining functions
you have to see what x cannot equal during the steps
finding the domains of function tables
you see what x values are included in both tables
absolute value function
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