Module 1 Flashcards
A Function:
Is a mapping from a set of inputs to a set of outputs with exactly one output for each input
If no domain is stated for a function y = f(x):
the domain is considered to be the set of all real numbers x for which the function is defined.
When sketching the graph of a function f
Each vertical line may intersect the graph, at most, once
How many zeros and y-intercepts does a function have?
A function may have any number of zeros but it will only have one y-intercept.
To define the composition g∘f:
the range of f must be contained in the domain of g.
Even Functions
are symmetric about the y-axis
Odd functions
Are symmetric about the origin
Composition of two functions
(g∘f)(x) = g(f(x))
Absolute value function
f(x) = |x| = {x,x≥0
{x,x<0
composite function
given two functions f and g, a new function, denoted g∘f, such that (g∘f)(x) = g(f(x)).
decreasing on the interval I
a function decreasing on the interval I if, for all x1 ,x2 ∈ I, f(x1) ≥ f(x2) if x1 < x2.
dependent variable
the output variable for a function
domain
the set of inputs for a function
even function
a function is even if f(−x) = f(x) for all x in the domain of f.
graph of a function
the set of points (x, y) such that x is in the domain of f and y=f(x)
increasing on the interval I
a function increasing on the interval I if for all x1,x2∈I,f(x1)≤f(x2) if x1 < x2
independent variable
the input variable for a function
odd function
a function is odd if f(−x)=−f(x) for all x in the domain of f
range
the set of outputs for a function
symmetry about the origin
the graph of a function f is symmetric about the origin if (−x,−y) is on the graph of f whenever (x,y) is on the graph.
symmetry about the y-axis
the graph of a function f is symmetric about the y-axis if (−x,y) is on the graph of f whenever (x,y) is on the graph
table of values
a table containing a list of inputs and their corresponding outputs
vertical line test
given the graph of a function, every vertical line intersects the graph, at most, once
zeros of a function
when a real number x is a zero of a function f, f(x) = 0
The power function f(x)=x^n is an even function or an odd function if
if n is even and n≠0, and it is an odd function if n is odd
What is the Domain of the root function f(x) = x^1/n?
has the domain [0,∞) if n is even and the domain (−∞,∞)
if n is odd.
What is the domain of the rational function f(x)=p(x)/q(x), where p(x) and q(x) are polynomial functions?
It is the set of x such that q(x)≠0.
{ x| q(x)≠0}
A polynomial function f with degree n≥1
satisfies f(x)→±∞ as x→±∞. The sign of the output as x→∞ depends on the sign of the leading coefficient only and on whether n is even or odd.
What are some examples of transformations of functions?
Vertical and horizontal shifts, vertical and horizontal scalings, and reflections about the x– and y-axes
Point-slope equation of a line
y − y1 = m(x − x1)
Slope-intercept form of a line
y = mx+b
Standard form of a line
ax + by = c
Polynomial function
f(x) = a↓n x^n + a↓n−1 x^n−1 + … + a↓1 x + a↓0
algebraic function
a function involving any combination of only the basic operations of addition, subtraction, multiplication, division, powers, and roots applied to an input variable
x
Transcendental function
Trigonometric, exponential, and logarithmic functions are examples of transcendental functions.
cubic function
a polynomial of degree 3; that is, a function of the form f(x) = ax^3+bx^2+cx+d, where a≠0
degree
for a polynomial function, the value of the largest exponent of any term
linear function
a function that can be written in the form f(x)=mx+b
logarithmic function
a function of the form f(x)=log↓b(x) for some base b>0,b≠1 such that y=log↓b(x) if and only if b^y=x
mathematical model
A method of simulating real-life situations with mathematical equations
piecewise-defined function
a function that is defined differently on different parts of its domain
point-slope equation
equation of a linear function indicating its slope and a point on the graph of the function
polynomial function
a function of the form f(x)=a↓n x^n + a↓n−1 x^n−1 + ⋯ + a↓1 x + a↓0
power function
a function of the form f(x)=x^n for any positive integer n≥1
quadratic function
a polynomial of degree 2; that is, a function of the form f(x)=ax^2+bx+c where a≠0
rational function
a function of the form f(x)=p(x)/qx), where p(x) and q(x) are polynomials
root function
a function of the form f(x)=x^1/n for any integer n≥2
slope
the change in y for each unit change in x
slope-intercept form
equation of a linear function indicating its slope and y-intercept
standard form
equation of a linear function with both variable terms set equal to a constant, ax+by=c.
transcendental function
a function that cannot be expressed by a combination of basic arithmetic operations
transformation of a function
a shift, scaling, or reflection of a function
Generalized sine function equation
f(x) = Asin(B(x−α)) + C
periodic function
a function is periodic if it has a repeating pattern as the values of x move from left to right
radians
for a circular arc of length s on a circle of radius 1, the radian measure of the associated angle θ is s
trigonometric functions
functions of an angle defined as ratios of the lengths of the sides of a right triangle
trigonometric identity
an equation involving trigonometric functions that is true for all angles θ for which the functions in the equation are defined
Reciprocal identities
tanθ = sinθ cotθ = cosθ cscθ = 1 secθ = 1
cosθ sinθ sinθ cosθ
Pythagorean identities
sin^2 θ + cos^2 θ = 1 1 + tan^2 θ = sec^2 θ
1 + cot^2 θ = csc^2 θ
Addition and subtraction formulas
sin(α±β) = sinα cosβ ± cosα sinβ cos(α±β) = cosα cosβ ∓ sinα sinβ
Double-angle formulas
sin(2θ) = 2sinθ cosθ cos(2θ) = 2cos^2 θ − 1 = 1 − 2sin^2 θ = cos^2 θ − sin^2 θ
Radian measure is
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 of 180° has a radian measure of π rad.
For acute angles 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 is θ.
For a general angle θ
let (x,y) be a point on a circle of radius r corresponding to this angle θ. The trigonometric functions can be written as ratios involving x,y, and r.
The trigonometric functions are periodic.
The sine, cosine, secant, and cosecant functions have period 2π. The tangent and cotangent functions have period π.
horizontal line test
a function f is one-to-one if and only if every horizontal line intersects the graph of f, at most, once
inverse function
for a function f, the inverse function f^−1 satisfies
f^−1 (y) = x if f(x) = y
inverse trigonometric functions
the inverses of the trigonometric functions are defined on restricted domains where they are one-to-one functions
one-to-one function
a function f is one-to-one if f(x1) ≠ f(x2) if x1 ≠ x2
restricted domain
a subset of the domain of a function f
When is the exponential function y=b^x increasing/decreasing and what is its domain and range
is increasing if b>1 and decreasing if 0<b></b>
The logarithmic function y=log↓b (x) is
is the inverse of y=b^x. Its domain is (0,∞) and its range is (−∞,∞)
The natural exponential function is
the natural logarithmic function is
The natural exponential function is y=e^x and the natural logarithmic function is y=lnx=log↓e x
Given an exponential function or logarithmic function in base a, we can make a change of base to convert this function to any base
b>0,b≠1. We typically convert to base e
.
The hyperbolic functions involve combinations of
f the exponential functions e^x and e^−x. As a result, the inverse hyperbolic functions involve the natural logarithm.
base
the number b in the exponential function f(x)=b^x and the logarithmic function f(x)=log↓b x
exponent
the value x in the expression b^x
hyperbolic functions
the functions denoted sinh,cosh,tanh,csch,sech, and coth, which involve certain combinations of e^x and e^−x
inverse hyperbolic functions
the inverses of the hyperbolic functions where cosh and sech are restricted to the domain [0,∞); each of these functions can be expressed in terms of a composition of the natural logarithm function and an algebraic function
natural exponential function
the function f(x)=e^x
number e
as m gets larger, the quantity (1+(1/m))^m gets closer to some real number; we define that real number to be e; the value of e is approximately 2.718282
