Math Definitions Flashcards
A _________ f is a rule that assigns to each value x in a set D a unique value denoted f(x).
function
The set D is the _________ of the function.
domain
The ______ is the set of all values of f(x) produced as x values over the entire domain.
range
The _______________ is the variable associated with the domain.
independent variable
The ______________ belongs to the range.
dependent variable
The _______ of a function f is the set of all points (x,y) in the xy-plane that satisfy the equation y=f(x).
graph
The ____________ of a function is the expression on which the function works.
argument
A line through any two points on a curve is called a _____________.
secant line
The slope of the secant line can be interpreted as _____________________ of f over the interval [a,x].
the average rate of change
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.
the y-axis
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.
the x-axis
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.
the origin
An_______ function f has the property that f (-x) = f (x) for all x in the domain.
even
An_______ function f has the property that f (-x) = -f (x) for all x in the domain.
odd
An nth degree polynomial can have as many as n real _________ which as the values of at which P (x) = 0
zeros or roots
A ______________ is a ratio of two polynomials, where the domain consists of all values of x such that the denominator does not equal 0.
rational function
Functions that have different definitions on different parts of their domain are called _________________.
piecewise functions
The ____________ of a number a, denoted | a |, is the distance from a to 0 on the real number line.
absolute value
The ______________, S (x), is the slope of the curve y = f (x) at the point (x, f (x)).
slope function
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.
area function
__________________ have the form f (x) = bˣ, where the base b ≠ 1 is a positive real number.
Exponential functions
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).
increasing
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).
decreasing
A function is ________ on I if it is either increasing or decreasing on I.
monotonic
Given a function f, its ________ (if it exists) is a function f⁻¹ such that whenever y = f (x), then f⁻¹ (y) =x.
inverse
A function f is _____________ on a domain D if each value of f (x) corresponds to exactly one value of x in D.
one-to-one
The ____________________ says that every horizontal line intersects the graph of a one-to-one function at most once.
horizontal line test
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.
logarithmic function
The _________________ is f (x) = eˣ which has the base e ≈ 2.7182…
natural exponential function
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.
standard position
_______________ have their values repeat over every interval of some fixed length.
Periodic functions
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.
period
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)
the limit of f (x) as x approaches a equals L
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 ________________________________.
the limit of f (x) as x approaches from the right equals L
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 ________________________________.
the limit of f (x) as x approaches from the left equals L
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 _________________________________.
the limit of f (x) as x approaches a is infinity
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 _________________________________.
the limit of f (x) as x approaches a is negative infinity.
If lim_x→a⁺ f (x) = ±∞, or lim_x→a⁻ f (x) = ±∞, the line x = a is called a ______________.
vertical asymptote
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 ______________________________.
the limit of f (x) as x approaches infinity is L
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 ______________________________.
the limit of f (x) as x approaches negative infinity is M
The line y = L and y = M are _______________.
horizontal asymptotes
If f (x) becomes arbitrarily large as x becomes arbitrarily large, then we write ____________.
lim_x→∞ f (x) = ∞
A function f is ___________ at a if lim_x→a f (x) = f (a).
continuous
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).
removable
A _______________ occurs when the left and right limits exist at a but are unequal.
jump discontinuity
An ______________ occurs when the function has a vertical asymptote at a.
infinite discontinuity
A function f is _________________________ at a if lim_x→a⁺ f (x) = f (a)
continuous from the right (right-continuity)
A function f is _________________________ at a if lim_x→b⁻ f (x) = f (b)
continuous from the left (left-continuity)
A function f is ________________________ if it is continuous at all points of I.
continuous on an interval I
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.
the limit of f (x) as x approaches a is L
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.
infinite limit
