Analysis Of Functions Definitions Flashcard

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A correspondence between two variables, and can be written as a set of ordered pairs (x,y). (When relations are written as ordered pairs, we say x is related to y.)

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Relation

A correspondence between two variables, and can be written as a set of ordered pairs (x,y). (When relations are written as ordered pairs, we say x is related to y.)

1 to 1 Function

A function f is said to be one-to-one if, for any choice of numbers x1 and x2, x1=/=x2, in the domain of f, then f(x1)=/=f(x2).

Cartesian Product

The cartesian product of two sets, A and B (written AxB), is the set of all ordered pairs whose first member is an element of A, and whose second member is an element of B.

Function

Let X and Y be two nonempty sets of real numbers(*). A function from X into Y is a rule or a correspondence that associates with each element of X a unique element of Y. (*The two sets X and Y can also be sets of complex numbers. In the broad definition, X and Y can be any two sets.)

Odd Function (definition)

A function is odd if for every number x in its domain the number -x is also in the domain and f(-x)=-f(x).

Odd Function (theorem)

A function is odd iff its graph is symmetric with respect to the origin.

Even Function (definition)

A function is even if for every number x in its domain the number -x is also in the domain and f(-x)=f(x)

Even Function (theorem)

A function is even iff its graph is symmetric with respect to the y-axis

Composition of a Function

Given two functions f and g, the composite function, denoted by fog, is defined by fog(x)=f(g(x)). The domain of fog is the set of all numbers x in the domain of g such that g(x) is in the domain of f.

Quadratic Function

A quadratic function is a function equivalent to one of the form y=a*x^2+b*x+c, where a, b, and c are real numbers and a=/=0.

Average Rate of Change

If c is in the domain of a function y=f(x), the average rate of change of f between c and x is defined as: average rate of change=delta y/delta x=(f(x)-f(c))/(x-c), x=/=c. (Also called the difference quotient of f at c)

Parabola Vertex

(-b/2a, f(-b/2a))

Number of Zeros

A polynomial function cannot have more zeros than its degree

Turning points

If f is a polynomial function of degree n, then f has at most n-1 turning points

Point-slope equation of a line

y-y1=m(x-x1)

Vertical Line equation

x=a

Horizontal Line equation

y=b

General equation of a line

Ax+By+C=0

Slope-intercept equation of a line

y=mx+b

Midpoint Formula

(x,y)=((x1+x2)/2,(y1+y2)/2)

Distance Formula

d=((x2-x1)^2+(y2-y1)^2)^(1/2)

Quadratic Formula

x=(-b+/-(b^2-4ac)^(1/2))/2a

Discriminant

b^2-4ac

Power Function of Degree n

A power function of degree n is a function of the form f(x)=a*x^n where a is a real number, a=/=0, and n>0 is an integer.

Increasing Interval

A function f is increasing on an open interval I if, for any choice of x1 and x2 in I, with x1

Decreasing Interval

A function f is decreasing on an open interval I if, for any choice of x1 and x2 in I, with x1f(x2).

Constant Interval

A function f is constant on an open interval I if, for any choice of x1 and x2 in I, with x1=/=x2, the values in f(x) are equal.

Factor Theorem

Let f be a polynomial function. Then x-c is a factor of f(x) iff f(c)=0

Zero of Multiplicity m of f

If (x-r)^m is a factor of polynomial f and (x-r)^(m+1) is not a factor of f, then r is called a zero of multiplicity m of f.

Zero/root of f

If f is a polynomial function and r is a real number for which f(r)=0, then r is called a (real) zero of f, or root of f. If r is a (real) zero of f, then 1) r is an x-intercept of the graph of f. 2) (x-r) is a factor of f

Parabola (definition)

The set of all points that are equidistant from a fixed line (D) and a fixed point (F) that is not on the line.

Vertical Line Test

A set of points in the xy-plane is the graph of a function iff a vertical line intersects the graph in at most one point.

Two lines are parallel iff what is true?

Two distinct non-vertical lines are parallel iff their slopes are equal.

Matrix

A rectangular array of numbers.

Two lines are perpendicular iff what is true?

Two non-vertical lines are perpendicular iff the product of their slopes is -1.

Local Maximum

A function f has a local maximum at c if there is an interval I containing c so that, for all x=/=c in I, f(x)

Local Minimum

A function f has a local minimum at c if there is an interval I containing c so that, for all x=/=c in I, f(x)>f(c). We call f(c) a local minimum of f.

Vertical Asymptotes (theorem)

A rational function R(x)=p(x)/q(x), in lowest terms, will have a vertical asymptote at x=r, if x-r is a factor of q(x).

Remainder Theorem

Let f be a polynomial function. If f(x) is divisible by x-c, then the result is f(c).

Fundamental Theorem of Algebra

Every complex polynomial function f(x) of degree n>=1 has at least 1 complex zero.

Complex Polynomial

A complex polynomial function f of degree n is a complex function of the form f(x)=an*x^n+an-1*x^(n-1)+...+a1*x+a0, where an, an-1,...,a1,a0 are complex numbers, an=/=0, n is a non-negative integer, and x is a complex variable.

Inverse of a Function

Let f denote a 1 to 1 function y=f(x). The inverse of f, denoted by f^-1, is a function such that f^-1(f(x))=x for every x in the domain of f and f(f^-1(x))=x for every x in the domain of f^-1.

Rational Zeros Theorem

Let f be a polynomial function of degree 1 or higher of the form f(x)=an*x^n+an-1*x^(n-1)+...+a1*x+a0, an=/=0, a0 =/=0 where each coefficient is an integer. If p/q in lowest terms is a rational zero of f, then p must be a factor of a0 and q must be a factor of an.

Polynomial Function

A polynomial function is a function of the form f(x)=an*x^n+an-1*x^(n-1)+...+a1*x+a0, where an, an-1,..., a1, a0 are real numbers and n is a non-negative integer. The domain consists of all real numbers.

Rational Function

A rational function is a function in the form R(x)=p(x)/q(x), where p and q are polynomial functions and q is not the zero polynomial.

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