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Fundamental Counting Principle : if one event can occur inmmways and a second event can occur innnways after the first event has occurred, then the two events can occur inmÃnmÃnways; also known as the Multiplication Principle | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
geometric sequence : a sequence in which the ratio of a term to a previous term is a constant | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
geometric series : the sum of the terms in a geometric sequence | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
index of summation : in summation notation, the variable used in the explicit formula for the terms of a series and written below the sigma with the lower limit of summation | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
infinite sequence : a function whose domain is the set of positive integers | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
infinite series : the sum of the terms in an infinite sequence | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
lower limit of summation : the number used in the explicit formula to find the first term in a series | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
Multiplication Principle : if one event can occur inmmways and a second event can occur innnways after the first event has occurred, then the two events can occur inmÃnmÃnways; also known as the Fundamental Counting Principle | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
mutually exclusive events : events that have no outcomes in common | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
n factorial : the product of all the positive integers from 1 tonn | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
nth partial sum : the sum of the firstnnterms of a sequence | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
nth term of a sequence : a formula for the general term of a sequence | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
outcomes : the possible results of an experiment | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
permutation : a selection of objects in which order matters | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
probability : a number from 0 to 1 indicating the likelihood of an event | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
probability model : a mathematical description of an experiment listing all possible outcomes and their associated probabilities | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
recursive formula : a formula that defines each term of a sequence using previous term(s) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
sample space : the set of all possible outcomes of an experiment | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
sequence : a function whose domain is a subset of the positive integers | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
series : the sum of the terms in a sequence | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
summation notation : a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
term : a number in a sequence | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
union of two events : the event that occurs if either or both events occur | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
upper limit of summation : the number used in the explicit formula to find the last term in a series | https://openstax.org/books/algebra-and-trigonometry-2e/pages/13-key-terms |
A function is a mapping from a set of inputs to a set of outputs with exactly one output for each input. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
If no domain is stated for a functiony=f(x),y=f(x),the domain is considered to be the set of all real numbersxxfor which the function is defined. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
When sketching the graph of a functionf,f,each vertical line may intersect the graph, at most, once. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
A function may have any number of zeros, but it has, at most, oney-intercept. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
To define the compositiongâf,gâf,the range offfmust be contained in the domain ofg.g. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
Even functions are symmetric about theyy-axis whereas odd functions are symmetric about the origin. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
The power functionf(x)=xnf(x)=xnis an even function ifnnis even andnâ0,nâ0,and it is an odd function ifnnis odd. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
The root functionf(x)=x1/nf(x)=x1/nhas the domain[0,â)[0,â)ifnnis even and the domain(ââ,â)(ââ,â)ifnnis odd. Ifnnis odd, thenf(x)=x1/nf(x)=x1/nis an odd function. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
The domain of the rational functionf(x)=p(x)/q(x),f(x)=p(x)/q(x),wherep(x)p(x)andq(x)q(x)are polynomial functions, is the set ofxxsuch thatq(x)â0.q(x)â0. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
Functions that involve the basic operations of addition, subtraction, multiplication, division, and powers are algebraic functions. All other functions are transcendental. Trigonometric, exponential, and logarithmic functions are examples of transcendental functions. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
A polynomial functionffwith degreenâ¥1nâ¥1satisfiesf(x)â±âf(x)â±âasxâ±â.xâ±â.The sign of the output asxââxââdepends on the sign of the leading coefficient only and on whethernnis even or odd. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
Vertical and horizontal shifts, vertical and horizontal scalings, and reflections about thexx- andyy-axes are examples of transformations of functions. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
Radian measure is defined such that the angle associated with the arc of length 1 on the unit circle has radian measure 1. An angle with a degree measure of180°180°has a radian measure ofÏÏrad. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
For acute anglesθ,θ,the values of the trigonometric functions are defined as ratios of two sides of a right triangle in which one of the acute angles isθ.θ. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
For a general angleθ,θ,let(x,y)(x,y)be a point on a circle of radiusrrcorresponding to this angleθ.θ.The trigonometric functions can be written as ratios involvingx,y,x,y,andr.r. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
The trigonometric functions are periodic. The sine, cosine, secant, and cosecant functions have period2Ï.2Ï.The tangent and cotangent functions have periodÏ.Ï. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
For a function to have an inverse, the function must be one-to-one. Given the graph of a function, we can determine whether the function is one-to-one by using the horizontal line test. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
If a function is not one-to-one, we can restrict the domain to a smaller domain where the function is one-to-one and then define the inverse of the function on the smaller domain. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
For a functionffand its inversefâ1,f(fâ1(x))=xfâ1,f(fâ1(x))=xfor allxxin the domain offâ1fâ1andfâ1(f(x))=xfâ1(f(x))=xfor allxxin the domain off.f. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
Since the trigonometric functions are periodic, we need to restrict their domains to define the inverse trigonometric functions. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
The graph of a functionffand its inversefâ1fâ1are symmetric about the liney=x.y=x. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
The exponential functiony=bxy=bxis increasing ifb>1b>1and decreasing if0<b<1.0<b<1.Its domain is(ââ,â)(ââ,â)and its range is(0,â).(0,â). | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
The logarithmic functiony=logb(x)y=logb(x)is the inverse ofy=bx.y=bx.Its domain is(0,â)(0,â)and its range is(ââ,â).(ââ,â). | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
The natural exponential function isy=exy=exand the natural logarithmic function isy=lnx=logex.y=lnx=logex. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
Given an exponential function or logarithmic function in basea,a,we can make a change of base to convert this function to any baseb>0,bâ1.b>0,bâ1.We typically convert to basee.e. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
The hyperbolic functions involve combinations of the exponential functionsexexandeâx.eâx.As a result, the inverse hyperbolic functions involve the natural logarithm. | https://openstax.org/books/calculus-volume-1/pages/1-key-concepts |
( g â f ) ( x ) = g ( f ( x ) ) ( g â f ) ( x ) = g ( f ( x ) ) | https://openstax.org/books/calculus-volume-1/pages/1-key-equations |
f ( x ) = { â x , x < 0 x , x ⥠0 f ( x ) = { â x , x < 0 x , x ⥠0 | https://openstax.org/books/calculus-volume-1/pages/1-key-equations |
y â y 1 = m ( x â x 1 ) y â y 1 = m ( x â x 1 ) | https://openstax.org/books/calculus-volume-1/pages/1-key-equations |
y = m x + b y = m x + b | https://openstax.org/books/calculus-volume-1/pages/1-key-equations |
a x + b y = c a x + b y = c | https://openstax.org/books/calculus-volume-1/pages/1-key-equations |
f ( x ) = a n x n + a n â 1 x n â 1 + ⯠+ a 1 x + a 0 f ( x ) = a n x n + a n â 1 x n â 1 + ⯠+ a 1 x + a 0 | https://openstax.org/books/calculus-volume-1/pages/1-key-equations |
f ( x ) = A sin ( B ( x â α ) ) + C f ( x ) = A sin ( B ( x â α ) ) + C | https://openstax.org/books/calculus-volume-1/pages/1-key-equations |
f â1 ( f ( x ) ) = x for all x in D , and f ( f â1 ( y ) ) = y for all y in R . f â1 ( f ( x ) ) = x for all x in D , and f ( f â1 ( y ) ) = y for all y in R . | https://openstax.org/books/calculus-volume-1/pages/1-key-equations |
absolute value function : f(x)={âx,x<0x,xâ¥0f(x)={âx,x<0x,xâ¥0 | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
algebraic function : a function involving any combination of only the basic operations of addition, subtraction, multiplication, division, powers, and roots applied to an input variablexx | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
base : the numberbbin the exponential functionf(x)=bxf(x)=bxand the logarithmic functionf(x)=logbxf(x)=logbx | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
composite function : given two functionsffandg,g,a new function, denotedgâf,gâf,such that(gâf)(x)=g(f(x))(gâf)(x)=g(f(x)) | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
cubic function : a polynomial of degree 3; that is, a function of the formf(x)=ax3+bx2+cx+d,f(x)=ax3+bx2+cx+d,whereaâ0aâ0 | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
decreasing on the intervalII : a function decreasing on the intervalIIif, for allx1,x2âI,f(x1)â¥f(x2)x1,x2âI,f(x1)â¥f(x2)ifx1<x2x1<x2 | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
degree : for a polynomial function, the value of the largest exponent of any term | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
dependent variable : the output variable for a function | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
domain : the set of inputs for a function | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
even function : a function is even iff(âx)=f(x)f(âx)=f(x)for allxxin the domain offf | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
exponent : the valuexxin the expressionbxbx | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
function : a set of inputs, a set of outputs, and a rule for mapping each input to exactly one output | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
graph of a function : the set of points(x,y)(x,y)such thatxxis in the domain offfandy=f(x)y=f(x) | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
horizontal line test : a functionffis one-to-one if and only if every horizontal line intersects the graph off,f,at most, once | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
hyperbolic functions : the functions denotedsinh,cosh,tanh,csch,sech,sinh,cosh,tanh,csch,sech,andcoth,coth,which involve certain combinations ofexexandeâxeâx | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
increasing on the intervalII : a function increasing on the intervalIIif for allx1,x2âI,f(x1)â¤f(x2)x1,x2âI,f(x1)â¤f(x2)ifx1<x2x1<x2 | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
independent variable : the input variable for a function | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
inverse function : for a functionf,f,the inverse functionfâ1fâ1satisfiesfâ1(y)=xfâ1(y)=xiff(x)=yf(x)=y | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
inverse hyperbolic functions : the inverses of the hyperbolic functions wherecoshcoshandsechsechare restricted to the domain[0,â);[0,â);each of these functions can be expressed in terms of a composition of the natural logarithm function and an algebraic function | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
inverse trigonometric functions : the inverses of the trigonometric functions are defined on restricted domains where they are one-to-one functions | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
linear function : a function that can be written in the formf(x)=mx+bf(x)=mx+b | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
logarithmic function : a function of the formf(x)=logb(x)f(x)=logb(x)for some baseb>0,bâ1b>0,bâ1such thaty=logb(x)y=logb(x)if and only ifby=xby=x | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
mathematical model : A method of simulating real-life situations with mathematical equations | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
natural exponential function : the functionf(x)=exf(x)=ex | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
natural logarithm : the functionlnx=logexlnx=logex | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
number e : asmmgets larger, the quantity(1+(1/m))m(1+(1/m))mgets closer to some real number; we define that real number to bee;e;the value ofeeis approximately2.7182822.718282 | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
odd function : a function is odd iff(âx)=âf(x)f(âx)=âf(x)for allxxin the domain offf | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
one-to-one function : a functionffis one-to-one iff(x1)âf(x2)f(x1)âf(x2)ifx1âx2x1âx2 | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
periodic function : a function is periodic if it has a repeating pattern as the values ofxxmove from left to right | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
piecewise-defined function : a function that is defined differently on different parts of its domain | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
point-slope equation : equation of a linear function indicating its slope and a point on the graph of the function | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
polynomial function : a function of the formf(x)=anxn+anâ1xnâ1+â¦+a1x+a0f(x)=anxn+anâ1xnâ1+â¦+a1x+a0 | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
power function : a function of the formf(x)=xnf(x)=xnfor any positive integernâ¥1nâ¥1 | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
quadratic function : a polynomial of degree 2; that is, a function of the formf(x)=ax2+bx+cf(x)=ax2+bx+cwhereaâ0aâ0 | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
radians : for a circular arc of lengthsson a circle of radius 1, the radian measure of the associated angleθθisss | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
range : the set of outputs for a function | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
rational function : a function of the formf(x)=p(x)/q(x),f(x)=p(x)/q(x),wherep(x)p(x)andq(x)q(x)are polynomials | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
restricted domain : a subset of the domain of a functionff | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
root function : a function of the formf(x)=x1/nf(x)=x1/nfor any integernâ¥2nâ¥2 | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
slope : the change inyfor each unit change inx | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
slope-intercept form : equation of a linear function indicating its slope andy-intercept | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
symmetry about the origin : the graph of a functionffis symmetric about the origin if(âx,ây)(âx,ây)is on the graph offfwhenever(x,y)(x,y)is on the graph | https://openstax.org/books/calculus-volume-1/pages/1-key-terms |
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