Math Definitions

Question 1

A _________ f is a rule that assigns to each value x in a set D a unique value denoted f(x).

Answer 1

function

Question 2

The set D is the _________ of the function.

Answer 2

domain

Question 3

The ______ is the set of all values of f(x) produced as x values over the entire domain.

Answer 3

range

Question 4

The _______________ is the variable associated with the domain.

Answer 4

independent variable

Question 5

The ______________ belongs to the range.

Answer 5

dependent variable

Question 6

The _______ of a function f is the set of all points (x,y) in the xy-plane that satisfy the equation y=f(x).

Answer 6

graph

Question 7

The ____________ of a function is the expression on which the function works.

Answer 7

argument

Question 8

A line through any two points on a curve is called a _____________.

Answer 8

secant line

Question 9

The slope of the secant line can be interpreted as _____________________ of f over the interval [a,x].

Answer 9

the average rate of change

Question 10

A graph is symmetric with respect to ________ if whenever the point (x,y) is on the graph, the points (-x,y) is also on the graph.

Answer 10

the y-axis

Question 11

A graph is symmetric with respect to ________ if whenever the point (x,y) is on the graph, the points (x,-y) is also on the graph.

Answer 11

the x-axis

Question 12

A graph is symmetric with respect to ________ if whenever the point (x,y) is on the graph, the points (-x,-y) is also on the graph.

Answer 12

the origin

Question 13

An_______ function f has the property that f (-x) = f (x) for all x in the domain.

Answer 13

even

Question 14

An_______ function f has the property that f (-x) = -f (x) for all x in the domain.

Answer 14

odd

Question 15

An nth degree polynomial can have as many as n real _________ which as the values of at which P (x) = 0

Answer 15

zeros or roots

Question 16

A ______________ is a ratio of two polynomials, where the domain consists of all values of x such that the denominator does not equal 0.

Answer 16

rational function

Question 17

Functions that have different definitions on different parts of their domain are called _________________.

Answer 17

piecewise functions

Question 18

The ____________ of a number a, denoted | a |, is the distance from a to 0 on the real number line.

Answer 18

absolute value

Question 19

The ______________, S (x), is the slope of the curve y = f (x) at the point (x, f (x)).

Answer 19

slope function

Question 20

The ____________, A (x), is the area of the region bounded by the graph of f and the t-axis from t = 0 to t = x.

Answer 20

area function

Question 21

__________________ have the form f (x) = bˣ, where the base b ≠ 1 is a positive real number.

Answer 21

Exponential functions

Question 22

Let I be an interval containing the points x₁ and x₂. A function f is __________ on I if x₁< x₂ implies f (x₁) < f (x₂).

Answer 22

increasing

Question 23

Let I be an interval containing the points x₁ and x₂. A function f is __________ on I if x₁< x₂ implies f (x₁) > f (x₂).

Answer 23

decreasing

Question 24

A function is ________ on I if it is either increasing or decreasing on I.

Answer 24

monotonic

Question 25

Given a function f, its ________ (if it exists) is a function f⁻¹ such that whenever y = f (x), then f⁻¹ (y) =x.

Answer 25

inverse

Question 26

A function f is _____________ on a domain D if each value of f (x) corresponds to exactly one value of x in D.

Answer 26

one-to-one

Question 27

The ____________________ says that every horizontal line intersects the graph of a one-to-one function at most once.

Answer 27

horizontal line test

Question 28

For any base b > 0, with b ≠ 1, the _______________ base b, denoted y = log_b (x) is the inverse of the exponential function y = bˣ.

Answer 28

logarithmic function

Question 29

The _________________ is f (x) = eˣ which has the base e ≈ 2.7182…

Answer 29

natural exponential function

Question 30

An angle θ is in ________________ if its initial side is on the positive x-axis and its terminal side is the line segment OP between the origin and P.

Answer 30

standard position

Question 31

_______________ have their values repeat over every interval of some fixed length.

Answer 31

Periodic functions

Question 32

A function f is said to be periodic if f (x + P) = f (x), for all x in the domain, where the ________ P is the smallest positive real number that has this property.

Answer 32

period

Question 33

Suppose the function f is defined for all x near a except possibly at a. If f (x) is arbitrarily close to L (as close to L as we like) for all x sufficiently close (but not equal to a, we write: lim_x →a f (x) = L and say _________________________________.
(Alternate notation: f (x) → L as x → a)

Answer 33

the limit of f (x) as x approaches a equals L

Question 34

Suppose f is defined for all x near a with x > a. If f (x) is arbitrarily close to L for all x sufficiently close to a with x > a, we write: lim_x→a⁺ f (x) = L and say ________________________________.

Answer 34

the limit of f (x) as x approaches from the right equals L

Question 35

Suppose f is defined for all x near a with x < a. If f (x) is arbitrarily close to L for all x sufficiently close to a with x < a, we write: lim_x→a⁻ f (x) = L and say ________________________________.

Answer 35

the limit of f (x) as x approaches from the left equals L

Question 36

Suppose f is defined for all x near a. If f (x) grows arbitrarily large for all x sufficiently close (but not equal) to a, we write: lim_x→a f (x) = ∞ and say _________________________________.

Answer 36

the limit of f (x) as x approaches a is infinity

Question 37

If f (x) is negative and grows arbitrarily large in magnitude for all x sufficiently close (but not equal) to a, we write: lim_x→a f (x) = -∞ and say _________________________________.

Answer 37

the limit of f (x) as x approaches a is negative infinity.

Question 38

If lim_x→a⁺ f (x) = ±∞, or lim_x→a⁻ f (x) = ±∞, the line x = a is called a ______________.

Answer 38

vertical asymptote

Question 39

If f (x) becomes arbitrarily close to a finite number L for all sufficiently large and positive x, then we write: lim_x→∞ f (x) = L and say ______________________________.

Answer 39

the limit of f (x) as x approaches infinity is L

Question 40

If f (x) becomes arbitrarily close to a finite number M for all sufficiently large in magnitude and negative x, then we write: lim_x→-∞ f (x) = M and say ______________________________.

Answer 40

the limit of f (x) as x approaches negative infinity is M

Question 41

The line y = L and y = M are _______________.

Answer 41

horizontal asymptotes

Question 42

If f (x) becomes arbitrarily large as x becomes arbitrarily large, then we write ____________.

Answer 42

lim_x→∞ f (x) = ∞

Question 43

A function f is ___________ at a if lim_x→a f (x) = f (a).

Answer 43

continuous

Question 44

A point of discontinuity, a, is called __________ if the function can be defined or redefined at a such that f (a) = lim_x→a f (x).

Answer 44

removable

Question 45

A _______________ occurs when the left and right limits exist at a but are unequal.

Answer 45

jump discontinuity

Question 46

An ______________ occurs when the function has a vertical asymptote at a.

Answer 46

infinite discontinuity

Question 47

A function f is _________________________ at a if lim_x→a⁺ f (x) = f (a)

Answer 47

continuous from the right (right-continuity)

Question 48

A function f is _________________________ at a if lim_x→b⁻ f (x) = f (b)

Answer 48

continuous from the left (left-continuity)

Question 49

A function f is ________________________ if it is continuous at all points of I.

Answer 49

continuous on an interval I

Question 50

Assume f (x) is defined for all x in some open interval containing a, except possibly at a, we say __________________________, written lim_x→a f (x) = L, if for any number ε > 0 there is a corresponding number δ > 0 such that |f (x) - L| < ε whenever 0 < |x - a| < δ.

Answer 50

the limit of f (x) as x approaches a is L

Question 51

The _________________ lim_x→a f (x) = ∞ means that for any positive number N, there exists a corresponding δ > 0 such that f (x) > N whenever 0 < |x - a| < δ.

Answer 51

infinite limit