Relations & Functions Flashcards
set
a well-defined collection of objects, which are called the elements of the set
empty set
∅ = {} = {x | x ≠ x}
set of natural numbers
N = {1, 2, 3, …}
set of whole numbers
W = {0, 1, 2, …}
set of integers
Z = {…, -3, -2, -1, 0, 1, 2, 3, …}
set of rational numbers
Q = {a/b | a ∈ Z and b ∈ Z}
set of real numbers
R = {x | x possesses a decimal representation}
set of irrational numbers
P = {x | x is a non-rational real number—that is, a decimal that does not repeat or terminate}
set of complex numbers
C = {a + bi | a,b ∈ R and i = √-1}
interval
the set of all real numbers lying between two fixed endpoints with no gaps
intersection of two sets
A ∩ B = {x | x ∈ A and x ∈ B}
union of two sets
A ∪ B = {x | x ∈ A or x ∈ B (or both}
Quadrant I
x > 0, y > 0
Quadrant II
x < 0, y > 0
Quadrant III
x < 0, y < 0
Quadrant IV
x > 0, y < 0
(a, b) and (c, d) are symmetric about the x-axis if
a = c and b = -d
(a, b) and (c, d) are symmetric about the y-axis if
a = -c and b = d
(a, b) and (c, d) are symmetric about the origin if
a = -c and b = -d
to reflect (x, y) about the x-axis,
replace y with -y
to reflect (x, y) about the y-axis,
replace x with -x
to reflect (x, y) about the origin,
replace x with -x and y with -y
the distance between points P(a, b) and Q(c, d) is
d = √[(c – a)² + (d – b)²]
the midpoint M of the line segment connecting P(a, b) and Q(c, d) is
M = [(a + c)/2, (b + d)/2]
relation
a set of points in the plane
the graph of x = a is
a vertical line through (a, 0)
the graph of y = b is
a horizontal line through (0, b)
the fundamental graphing principle
The graph of an equation is the set of points which satisfy the equation. That is, a point (x, y) is on the graph if and only if x and y satisfy the equation.
x-intercept
a point on a graph which is also on the x-axis
y-intercept
a point on the graph which is also on the y-axis
how to find an x-intercept
set y = 0 and solve for x to find the x-intercepts (x, 0)
how to find a y-intercept
set x = 0 and solve for y to find the y-intercepts (0, y)
test the graph of an equation for symmetry about the y-axis
substitute (–x, y) and simplify. The graph is symmetric about the y-axis if the result is equivalent to the original equation
test the graph of an equation for symmetry about the x-axis
substitute (x, –y) and simplify. The graph is symmetric about the x-axis if the result is equivalent to the original equation
test the graph of an equation for symmetry about the origin
substitute (–x, –y) and simplify. The graph is symmetric about the origin if the result is equivalent to the original equation
function (definition)
a rule that associates a unique output with each input
the vertical line test
a set of points in the plane represents y as a function of x if and only if no two points lie on the same vertical line
domain (definition)
the set of all allowable inputs (x-values) of a function
range (definition)
the set of outputs (y-values) that result when x varies over the domain of a function
how to determine whether an equation represents y as a function of x
solve for y and determine whether each choice of x will determine only one corresponding value of y
independent variable
x is the independent variable (or argument) of f
dependent variable
y is the dependent variable of f
difference quotient
[f(x+h) – f(x)]/h
the fundamental graphing principle for functions
The graph of a function f is the set of points which satisfy the equation y = f(x). That is, the point (x, y) is on the graph of f if and only if y = f(x).
zeros (definition)
the values of x for which f(x) = 0
the x-coordinates of the points where the graph of f intersects the x-axis
also called roots or x-intercepts
the graph of a function f is symmetric about the y-axis
if and only if f(–x) = f(x) for all x in the domain of f
the graph of a function f is symmetric about the origin
if and only if f(–x) = –f(x) for all x in the domain of f
even function
a function whose graph is symmetric about the y-axis
odd function
a function whose graph is symmetric about the origin
a function f is increasing on an interval I if
f(a) < f(b) for all real numbers a, b in I with a < b
a function f is decreasing on an interval I if and only if
f(a) > f(b) for all real numbers a, b in I with a < b
a function f is constant on an interval I if and only if
f(a) = f(b) for all real numbers a, b in I
Suppose f is a function with f(a) = b. f has a local maximum at (a, b) if and only if
there is an open interval I containing a for which f(a) ≥ f(x) for all x in I
Suppose f is a function with f(a) = b. f has a local minimum at the point (a, b) if and only if
there is an open interval I containing a for which f(a) ≤ f(x) for all x in I
f(a) = b is a local minimum value of f in this case
Suppose f(a) = b. The value b is a maximum of f if
b ≥ f(x) for all x in the domain of f
Suppose f(a) = b. The value b is called the minimum of f if
b ≤ f(x) for all x in the domain of f
Suppose k is a positive number. To graph y = f(x) + k,
shift the graph up k units by adding k to the y-coordinates of the points on the graph of f
Suppose k is a positive number. To graph y = f(x) – k,
shift the graph of y = f(x) down k units by subtracting k from the y-coordinates of the points on the graph of f
how to shift a function vertically
adding to or subtracting from the output of a function causes the graph to shift up or down, respectively
add k to the y-coordinates to shift f up k units: y = f(x) + k
subtract k from the y-coordinates to shift y down k units: y = f(x) – k
Suppose h is a positive number. To graph y = f(x + h),
shift the graph of y = f(x) left h units by subtracting h from the x-coordinates of the points on the graph of f
Suppose h is a positive number. To graph y = f(x – h),
shift the graph of y = f(x) right h units by adding h to the x-coordinates of the points on the graph of f
how to shift a graph horizontally
adding to or subtracting from the input of a function amounts to shifting the graph left or right, respectively
To graph y = –f(x),
reflect the graph of y = f(x) across the x-axis by multiplying the y-coordinates of the points on the graph of f by -1
To graph y = f(–x),
reflect the graph of y = f(x) across the y-axis by multiplying the x-coordinates of the points on the graph of f by –1
how to reflect a graph across the x-axis
multiply the output from a function [the y-coordinates] by –1
how to reflect a graph across the y-axis
multiply the input [the x-coordinates] to the function by -1
rigid transformations (definition)
vertical shifts, horizontal shifts, and reflections
these transformations change the position and orientation of a graph in the plane, but not its shape
non-rigid transformations (definition)
transformations, including vertical and horizontal scalings, that change the shape of the graph
Suppose a > 0. To graph y = af(x),
multiply all of the y-coordinates of the points on the graph of f by a
how to stretch a graph vertically
multiply all of the y-coordinates of the points by a factor of a, a > 1
how to shrink a graph vertically
multiply all of the y coordinates on the points of the graph by a, 0 < a < 1
Suppose b > 0. To graph y = f(bx),
divide all of the x-coordinates of the points on the graph of f by b
we say that f has been horizontally scaled by a factor of 1/b
how to stretch a graph horizontally
divide all of the x-coordinates of the points of the graph by b, 0 < b < 1
y = f(bx), f stretches horizontally by a factor of 1/b
how to shrink a graph horizontally
divide all of the x-coordinates of the points on the graph of f by b, b > 1
if b > 1, y = f(bx) and f shrinks horizontally by a factor of b
how to graph g(x) = Af(Bx + H) + K
- subtract H from each x-coordinate, shifting f to the left if H > 0 or right if H < 0
- divide the (new) x-coordinates by B, resulting in a horizontal scaling (and a reflection about the y-axis if B > 0)
- multiply the y-coordinates by A, resulting in a vertical scaling (and a reflection about the x-axis if A < 0)
- add K to each of the (new) y-coordinates, resulting in a vertical shift up if K > 0 or down if K < 0
real-valued function of a real variable (definition)
a function where the independent and dependent variables are real numbers
natural domain (definition)
where the domain for a real-valued function of a real variable is not explicitly stated, then the natural domain consists of all real numbers for which the formula yields a real value
