Chp 1 Flashcards
(90 cards)
1
Q
(a,b) = {x | a < x < b}
A
The open interval from A to B:
Notice that the endpoints of the interval—namely, a and b—are excluded. This is indicated by the round brackets and by open dots in a number line.
2
Q
[a,b] = {x | a <= x <= b}
A
The closed interval from a to b:
This is indicated by the square brackets and by closed dots in a number line.
3
Q
A function f is a rule that assigns to each element x in a set D
A
exactly one element, called f(x), in a set E.
4
Q
A function f is a rule that assigns to each element x in a set D
A
exactly one element, called f(x), in a set E.
5
Q
D =
A
Domain = X = Independent variable = all possible inputs
6
Q
E =
A
Range = Y = f(x) = Dependent = all possible outputs
7
Q
F(x)
A
F at x stated as F of x
8
Q
Function
A
One number depends on another
9
Q
given a number (r, t, w, or t), another number (A, P, C, or a) is assigned. In each case we say that …
A
the second number is a function of the first number. A function f is a rule that assigns to each element x in a set D exactly one element, called f(x), in a set E.
10
Q
in a graph of f, f(x) helps indicate visually the…
A
Height of the graph above point X
11
Q
In a graph of f, you can easily visualize…
A
The domain (width) and range (height) of the graph.
12
Q
It’s helpful to think of functions as machines where…
A
The Domain contains all possible INPUTS that always result in one OUTPUT from the Range
13
Q
P(t) is asking for
A
P at t
14
Q
P(t) means that
A
P is a function of t
15
Q
S U T
A
set S union set T (all elements of both)
16
Q
S 冂 T
A
Intersection (shared) elements between S and T.
17
Q
When asked to evaluate using a function, this just means…
A
Plug the expression into the function. e.g., plug in for x in f(x)
18
Q
x € with a slash through it… S
A
x is not a part of set S
19
Q
|
A
Means: Restricted to
20
Q
x € D
A
Restricted to x is an element of set D
21
Q
There are four possible ways to represent a function:
A
● verbally (by a description in words)
● numerically (by a table of values)
● visually (by a graph)
● algebraically (by an explicit formula)
22
Q
How can you tell if a curve represents a function on the XY plane?
A
The Vertical Line Test A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once.
23
Q
piecewise defined functions
A
defined by different formulas in different parts of their domains.
F(x) = {1-x If x <= -1
X^2 if x > -1}
24
Q
Point-slope form of the equation of a line:
A
y - y1 = m(x - x1)
y1 and x1 are from any point coordinates on the line. M is the slope.
So if you end up with
Y = 2 - x
Then
f(x) = 2 - x
25
Symmetry
| Even function
f satisfies f(-x) = f(x) for all x in its domain.
e.g. f(x) = x^2
f(-x) = (-x)^2 = x^2 = f(x)
NOTE: This doesn’t mean f(x) is always positive! This has nothing to do with Y or f(x) being positive!
It means that Y is always the same for negative and positive X values!
26
Symmetry
| Odd Function
f satisfies f(-x) = -f(x) for all x in its domain.
e.g. f(x) = x^3
f(-x) = (-x)^3 = -x^3 = -f(x)
27
Special property of even functions
Y is a mirror image greater than and less than 0
That is, it’s symmetrical about the Y axis.
28
Special property of odd functions
The graph of an odd function is symmetric about the origin
If we already have the graph of f for x > 0, we can obtain the entire graph by rotating this portion through 180 degrees about the origin.
29
Interpolation
Estimating a value within observed values.
30
extrapolation
Predicting a value outside the time frame of observations.
31
Function notation
e.g.:
f(x) = x^2 - 2
f(x) = x^2 - 2
f = function name
x = input
x^2 - 2 = output = f(x)
32
Integer
Whole number like 0, -1, 1 , 125
33
Rational number
Q
Natural numbers (non-negative)
Integers
Fractions of integers where denominator is not 0 (which is undefined)
Decimal pattern repeats or terminate (0 repeats)
E.g.:
9/7 = 1.285714285714... − 1.285714 repeats infinitely with a bar over it to represent that.
1/2 = 0.50000 = 0.50 with bar over 0
34
R
The set of all real numbers is usually denoted by the symbol R. When we use the word number without qualification, we mean “real number.”
35
Irrational numbers
Can’t be expressed with a fraction of integers
Does not have a repeating decimal pattern
E.g, Pie = 3.141592653589793...
36
Real Numbers Include
Rational, Irrational anything on an infinite number line.
37
Polynomial notation
f(x) = 2x^3 + 2x^2 - 1X - 3
Numbers: constants = coefficients
Exponents: non-negative integers
Lead Exponent: degree of the polynomial = 3
38
Quadratic Function
Form: f(x) = ax^2 + bX + c
Its graph is always a parabola obtained by shifting the parabola y − ax^2 as we will see in the next section. The parabola opens upward if a > 0 and downward if a < 0.
39
Cubic function
P(x)− ax^3 + bx^2 + cx + d
40
Quadratic formula
Used to solve for x in a quadratic formula:
Where ax^2 + bx + c = 0
X = -b +- sqrt(b^2 - 4ac)
——————————
2a
One negative boy couldn’t decide on going to a party or not because he was square. He missed out on meeting two awesome chicks. The party was over at 2a.
41
Power function
A function of the form f(x) = x^a, where a is a constant, is called a power function.
X is the base!
42
F(x) = x^2
An even power function
A parabola which flattens and steapens as n grows (assuming n is a multiple of , ie even)
43
f(x) = x^3
An odd power function
A split parabola (up on right side, down on left) which flattens and steapens as n grows (assuming n is odd , ie function is odd)
44
Rational function
rational function f is a ratio of two polynomials:
45
Algebraic function
algebraic function if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) starting with polynomials. Any rational function is automatically an algebraic function.
46
Trigonometric functions
| f(x) = sin x
the sine of the angle whose radian measure is x.
In calculus the convention is that radian measure is always used (except when otherwise indicated). For example, when we use the function f(x) = sin x, it is understood that sin x means the sine of the angle whose radian measure is x.
47
Key properties of trigonometric functions
Notice that for both the sine and cosine functions the domain is (-inf,Inf) and the range is the closed interval [-1,1].
Thus, for all values of x, we have -1 <= sin x < =1 -1 <= cos x < =1 or, in terms of absolute values, |sin x | < =1 | cos x | < =1
Also, the zeros of the sine function occur at the integer multiples of pie that is, sin x = 0 when x = n pie n an integer
An important property of the sine and cosine functions is that they are periodic functions and have period 2pie. This means that, for all values of x, sin(x + 2pi) = sin x and cos(x + 2pi) = cos x
The periodic nature of these functions makes them suitable for modeling repetitive phenomena such as tides, vibrating springs, and sound waves.
48
The domain of a composite function
Any input I am successfully feed through all functions individually and collectively in the composite.
49
Composite functions calculation
Put the second function into the first function (plug it in)
50
Classify the function:
| f(x) = x^5
f(x) = x^5 is a power function. (The x is the base.) We could also consider it to be a polynomial of degree 5.
51
Classify the function:
| f(x) = 5^x
f(x) = 5^x is an exponential function. (The x is the exponent.)
52
Classify the function:
| f(x) = 1 + x / 1 - sqrt(x)
An algebraic function
53
Classify the function:
| f(x) = 1 - t + 5t^4
Is a polynomial of degree 4
54
Inverse function
Swap the X and y and solve for y.
55
Inverse function graph
Reflects over X = Y
56
y = f(x) + c
| Where c > 0
Shift the graph of y = f(x) up c units
57
y = f(x) - c
| Where c > 0
Shift the graph of y = f(x) down c units
58
y = f(x + c)
| Where c > 0
Shift the graph of y = f(x) c units left
59
y = f(x - c)
| Where c > 0
Shift the graph of y = f(x) c units right
60
y = cf(x)
| Where c > 0
Stretch the graph of y = f(x) vertically by a factor of c units
61
y = (1/c)f(x)
| Where c > 0
Shrink (compress) the graph of y = f(x) vertically by a factor of c units
62
y = f(cx)
Shrink (compress) the graph of y = f(x) horizontally by a factor of c units
63
y = f(x/c)
Stretch the graph of f(x) by a factor of c horizontally.
64
y = -f(x)
Reflect the graph of y = f(x) about the x-axis
65
y = f(-x)
Reflect the graph of y = f(x) about the y-axis
66
Function transformations in the parentheses typically...
Moves left or right / horizontally and does the opposite of what’s expected.
67
Composition function
Given two functions f and g, the composite function f(g(x)) (also called the composition of f and g) is defined by (f o g)(x) = f(g(x))
68
Composite function machine
f o g (x)
| Plug x into g plug into f
69
Find equation of a line using point intercept form
y - y1 = m(x - x1)
| If given only a point and slope, plug in the slope then use (x,y) of the point to plug in and solve for b.
70
two restrictions on the domain:
1. A denominator cannot equal zero.
| 2. An algebraic expression under a square root sign must be positive or zero.
71
Properties of y = b^x
Exponential Function (constant is an exponent)
Upward sloping to right
x>=0
y intercept is always 1
Larger x steeper slope
y(0) = 1 (straight line)
Smaller 0 < x < 1 (fraction) reflects and gets steeper
-> since (1/b)^x = 1/b^x = b^-x, the graph y = (1/b)^x is just the reflection of y = b^x
about the y axis
72
Decimal Multiplication Rule
The number of decimals (places) of the numbers being multiplied sum to the number of decimals of the product.
73
b^x+y
b^x * b^y
74
b^x-y
b^x / b^y
75
(b^x)^y
b^(x*y)
76
(ab)^x
a^x * b^x
77
Flip the sign of the exponent by...
Taking its reciprocal
78
Determine if a function is odd, even or neither (Short cut!)
Constants (numbers) don’t matter, only variables and their exponents.
```
Even = all exponents are even
Odd = all exponents are odd (including no exponent = 1)
Neither = exponents are mixed odd / even
```
79
Determine if a function is odd, even or neither (formal)
Constants (numbers) don’t matter, only variables and their exponents.
Plug in negative sign for inputs and solve to see how it compares to the original f(x):
Even: if f(-x) = f(x)
Odd: if f(-x) = -f(x)
Neither: if f(-x) <> f(x) or -f(x)
80
f(x) = e^x
The natural exponential function
The. number e is the value where an exponential function’s y intercept (0,1) equals exactly 1.
e = 2.71828...
81
Complete the square (make a perfect square):
(X^2 + 4x)
(y^2 - 4y)
1) Square half of the second term
2) Add to both sides
3) factor
(X^2 + 4x) = b add 2 ^2
X^2 + 4x + 4 = b + 4 (add to both sides)
(X + 2)^2 = b
(y^2 - 4y) = b add -2^2
y^2 -4y +4 = b + 4 (add to both sides)
(y - 2)^2 = b + 4
82
Equation of a circle
hint: it’s just the pythagorean theorem.
An equation of the circle with center (h, k) and radius r is
(x - h)^2 + (y - k)^2 = r^2
In particular, if the center is the origin (0, 0), the equation is
x^2 + y^2 = r^2
83
f(x) = log sub2 x
Is the inverse of f(x) = 2^x
84
Sketch the graph equation:
x^2 +y^2 + 2x - 6y + 7 = 0
by first showing that it represents a circle and then finding its center and radius.
1) group the x and y-terms:
(X^2 + 2x) + (y^2 - 6y) = -7
2) Complete the square within each grouping adding the appropriate constants (squares of half the coefficients of x and y) to both sides of the equation:
(X^2 + 2x + 1) + (y^2 - 6y + 9) = -7 + 1 + 9
(X + 1)^2 + (y - 3)^2 = 3
Therefore:
Center = (1,-3)
r = sqrt(3)
85
Equation of an Ellipses
this is basically an oval:
(x^2 / a^2) + (y^2 / b^2) = 1
86
Equation of Hyperbolas
(x^2/a^2) - (y^2/b^2) =1
or
((x - h)^2/a^2) - ((y-k)^2/b^2) =1
87
Difference quotient
f(a + h) - f(a) / h
This represents the avg rate of change of f(x) between x = a and x = a + h
88
Exponents with negative bases
Follow order of operation
(-2)^2 = -2 * -2 because you do what’s in parentheses first which is -1 * 2 then exponent = 4
-2^2 = -1 * 2^2 = -4
Here, exponent goes first then multiply by -1
89
When simplifying fractions of fractions
1) always add the “/1” to whole numbers
2) Use GCF to add and subtract
3) Flip (reciprocal) the denominator and multiply the numerator is the same as dividing
4) MULTIPLYING ELEMENTS CAN MOVE LEFT AND RIGHT FREELY:
5) Structure your elements for cancelation / simplification
6) Remember to factor out elements to further look for cancelation matches
7) Anything kinked with “+ or -“ are one unit and must be factored or LCD or used as a whole.
8) LCD must be for the top and bottom denominators. What’s done to the numerator must be done to the denominator.
90
Tips for graphing functions
1) Make a table for x and y [f(x)]
2) Find Y and X intercepts
3) Check fractional inputs especially for piecewise functions and especially close to the origin
