MST124 BOOK A UNIT 3

- empty set is denoted by the symbol ∅
- the symbol for 'is in' is ∈
- the symbol for 'is not in' is ∈

- if A and B are any two sets, you can form a new set whose members are all the elements that belong to both A and B
- the intersection of A and B, denoted by A ∩ B

- if A and B are any two sets, then a new set can be formed whose members are all the objects that belong to either A or B or both
- union of A and B, denoted by A ∪ B

- sometimes, every element of set A is also an element of set B
- A is a subset of B, denoted by A ⊆ B
- every set is a subset of itself, and the empty set is a subset of every set 

- the set of all real numbers is denoted by R

- a solid dot on a number line indicates that it is included in the set, and a hollow dot indicates a number that isn't included 

- an interval that includes all of its endpoints is said to be closed
- one that doesn't include any of its endpoints is said to be open
- an interval that includes one endpoint but not the other is said to be half-open or half-closed 

Inequality signs
< is less than
≤ is less than or equal to
> is greater than
≥ is greater than or equal to 

- in interval notation, a [square bracket] indicates an included endpoint, and a (round bracket) indicates an excluded endpoint
- an interval that extends indefinitely is denoted using an infinity symbol, or -infinity 

- the rule of a function takes input values, processes them, and produces an output value 

A function consists of:
- a set of allowed input values, called the domain of the function
- a set of values in which every output value lies, called the codomain of the function
- a process, called the rule of the function, for converting each input value into exactly one output value 

- every value in the domain of a function must have a corresponding output value
- not every value in the codomain of a function actually has to occur as an output value 
- the image set of a function consists of the values in the codomain of a function that do occur as an output

Domain convention
- when just a rule specifies a function, it's understood that the domain of the function is the largest possible set of real numbers for which the rule is applicable 

- sometimes a function is expressed as an equation that expresses one variable in terms of another variable 
- the output variable is the dependent variable, and the input variable is the independent variable 

Piecewise-defined function
- a function can be specified by using different formulas for different parts of its domain
f(x) = {x² (x≥0), x+5 (x<0)}

- the graph of a function f is the graph of the equation y = f(x) for all the values of x that are in the domain of f 
- the input numbers are on the horizontal axis, and the output numbers are on the vertical axis 

Functions increasing or decreasing on an interval
A function f is increasing on the interval I if for all values x₁ and x₂ in I such that x₁ < x₂
f(x₁) < f(x₂)
A function f is decreasing on the interval I if for all values x₁ and x₂ in I such that x₁<x₂
f(x₁) > f(x₂) 

- any function of the form f(x) = mx+c is a linear function (m and c are constants)
- a linear function whose rule is of the form f(x) = c (c is a constant) is called a constant function 
- any function whose rule is of the form f(x) = ax² + bx + c is called a quadratic function 

- if an expression is a sum of finitely many terms, each of which is of the form axⁿ (where a is a number and n is a non-negative integer) then the expression is called a polynomial expression in x
- if the RHS of the rule of a function is a polynomial expression in x, then the function is called a polynomial function 

- the degree of the polynomial expression or function is the highest power of the variable x
- if the highest power of x in the rule is x⁴ then the degree of the polynomial function is 4 

- the graph of every polynomial function (with domain R) that isn't a constant function tends to infinity or minus infinity at the left and right 

- the dominant term in a rule is the highest power of x
- if the dominant term has a plus sign as contains an even power of x, then the graph tends to infinity at both ends
- if it is positive and contains an odd power of x, the graph tends to minus infinity at the left and to infinity at the right
- if the dominant term has a minus sign and an even power of x, the graph tends to minus infinity at both ends
- if it is negative and an odd power of x, the graph tends to infinity at the left and minus infinity at the right 

- the modulus function has a graph that's smooth except at one point where it turns a corner 
- the graph of the modulus function is the same as the graph of y = x when x ≥ 0, and the same as the graph of y = -x when x<0
- it has a corner at the origin and the image set is [0, infinity)

- if x is any non-zero number, then the reciprocal of x is 1/x
- the reciprocal function is the function f(x) = 1/x
- its domain consists of all real numbers except 0
(-infinity,0)U(0, infinity)

- the reciprocal function and all polynomial functions are examples of rational functions
- a rational function has the rule f(x) = p(x) / q(x) where p and q are polynomial functions 
- if q is a constant function, then f is a polynomial function
- if p(x) = 1 and q(x) = x then f is the reciprocal function 

- if a curve has the property that, as you trace your pen tip along it further and further from the origin, it gets arbitrarily close to a straight line, then that line is called an asymptote of the curve
- no matter how small a distance you choose, if you trace your pen top along the curve far enough, then eventually the curve lies within that distance of the line and stays within that distance of the line 

- every rational function has a graph that consists of one or more pieces, each with a smooth curve 

- the graphs of many rational functions have asymptotes, which can be horizontal, vertical, or slant 

Translations of graphs
Suppose that f is a function and c is a constant. To obtain the graph of:
y = f(x) + c, translate the graph of y = f(x) up by c units (translation is down if c is negative)
y = f(x-c), translate the graph of y = f(x) to the right by c units (translation if to the left if c is negative, so the equation would be f(x-(-c)) which is f(x-c))

- when you multiply the RHS of a function rule by a constant c, the new function obtained is called a constant multiple of the original function
- if c is +ve, then move each point on the graph vertically, in the direction away from the x-axis, until it's c times as far from the x-axis as it was before
- if c is -ve, then move each point on the graph vertically, in the direction away from the x-axis until is modulus c times as far from the x-axis as it was before, and then reflect it in the x-axis
- if c is 0, then move each point on the graph vertically until it lies on the x-axis 
- if the modulus of c is less than 1, each point is actually moved closer to the x-axis than it was before 

- in general, you can make any number of successive changes to the rule of a function to scale and translate its graph in various ways, but you have to be careful about the order in which you carry out the changes 
- sometimes the order does matter and sometimes it doesn't

- you can also scale a graph horizontally by a factor of c by changing the input variable in the RHS to x/c 
- if c is +ve, then move each point on the graph horizontally, in the direction away from the y-axis, until its c times as far from the y-axis as it was before 
- if c is -ve, then move each point horizontally, in the direction away from the y-axis, until its modulus c times as far from the y-axis as it was before, and then reflect it in the y-axis 
- if c is 0, then move each point on the graph horizontally until it lies on the y-axis 
- if the modulus of c is less than one, then each point is actually moved closer to the y-axis than it was before 

- to obtain the graph of y = -f(x), reflect the graph of y = f(x) in the x-axis
- to obtain the graph of y = f(-x), reflect the graph of y = f(x) in the y-axis 

- suppose that f and g are functions, the sum of f and g has the rule h(x) = f(x) + g(x)
- there are two differences of f and g, with rules h(x) = f(x) - g(x) and h(x) = g(x) - f(x)
- the product of f and g has the rule h(x) = f(x)*g(x)
- there are two quotients of f and g, with rules h(x) = f(x)/g(x) and h(x) = g(x)/f(x) 

- the domain of each of the combined functions above is the intersection of the domain of f and the domain of g
- for the quotients of the functions, neither one should have a denominator which equals 0

Composite functions
Suppose that f and g are functions. The composite function g o f is the function whose rule is (g o f)(x) = g(f(x))
and whose domain consists of all the values x in the domain of f such that f(x) is in the domain of g
- for this example, you would process x using the f function, and then that output would be the input for the g function 

- you can compose more than two functions
- e.g., (h o g o f)(x) = h(g(f(x)))

- the inverse of the function f (denoted by f^-1) is the function whose mapping diagram is obtained by reversing the directions of all the arrows in this full version of the diagram 
- if inputting a number x to give f gives the number y, then inputting the number y to f^-1 gives the original number x 

- some functions don't have inverse functions 
- this is because an input will have more than one output, which is not possible for a function
- functions only have one output per input 

One-to-one functions
A function f is one-to-one if for all values x₁ and x₂ in its domain such that x₁ does not equal x₂ 
f(x₁) does not equal f(x₂) 

- only one-to-one functions have inverse functions 
- functions that have an inverse are called invertible 

Inverse functions
Suppose that f is a one-to-one function, with domain A and image set B. Then the inverse function of f (denoted by f^-1) is the function with domain B whose rule is given by
f^-1(y) = x, where f(x) = y
The image set of f^-1 is A 

Strategy
To find the rule of the inverse function of a one-to-one function f 
- write y = f(x) and rearrange this equation to express x in terms of y
- use the resulting equation x = f^-1(y) to write down the rule of f^-1 
- usually change the input variable from y to x 

- if a function is either increasing on its whole domain, or decreasing on its whole domain, then it is one-to-one and so has an inverse function 

for any pair of inverse functions f and f^-1 
(f^-1 o f)(x) = x, for every value x in the domain of f
(f o f^-1)(x) = x, for every value x in the domain of f^-1 

- graphs of f and f^-1 are reflections of each other in the line y = x 

- starting with the function f that's not one-to-one, specify a new function that has the same rule as f but a smaller domain
- we choose the new domain to make sure that the following two conditions are satisfied:
1 - the new function is one-to-one and therefore has an inverse
2 - the image set of the new function is the same as the image set of the original function 

- a function that's obtained from another function f by keeping the rule the same but removing some numbers from the domain is called a restriction of the original function f
- the process of obtaining such a function is called restricting the domain of f, or restricting f 

- an exponential function is a function whose rule is of the form f(x) = b^x where b is a +ve constant, not equal to 1 

Graphs of exponential functions
The graph of the function f(x) = b^x, where b>0 and doesn't equal 1, has the following features 
- the graph lies entirely above the x-axis
- if b > 1, then the graph is increasing, and it gets steeper as x increases
- if 0<b<1, then the graph is decreasing, and it gets less steep as x increases
- the x-axis is an asymptote
- the y-intercept is 1
- the closer the value of b is to 1, the flatter the graph

- the value of b that gives a gradient of 1 at (0,1) is e (2.71828...)
- the number e is irrational
- the exponential function with the rule f(x) = e^x has the special property that its gradient is exactly 1 and the point (0,1)

Logarithms
The logarithm to base b of a number x, denoted by log_bx, is the power to which the base b must be raised to give the number x
y = log_bx and x = b^y are equal 
- the base b must be positive and not equal to 1
- only positive numbers have logarithms, but logarithms themselves can be any number 

- for any base b, log_b1 = 0 and log_bb = 1

Natural Logarithms
The natural logarithm of a number x, denoted by lnx, is the power to which the base e must be raised to give the number x. The two equations
y = lnx and x = e^y 
are equivalent 

ln1 = 0 and lne = 1 

Graphs of logarithmic functions
The graph of the function f(x) = log_bx, where b > 0 and b doesn't equal 1, has the following features 
- the graph lies entirely to the right of the y-axis
- if b > 1, then the graph is increasing and it gets less steep as x increases
- if 0 < b < 1 then the graph is decreasing and it gets less steep as x increases
- the y-axis is an asymptote 
- the x-intercept is 1
- the closer the value of b is to 1, the steeper the graph 

For any base b, 
log_b(b^x) = x and b^log_bx = x 
ln(e^x) = x and e^lnx = x 

Three logarithm laws
log_bx + log_by = log_b(xy)
log_bx - log_by = log_b(x/y)
rlog_bx = log_b(x^r) 

ln(x^r) = r lnx

- any exponential function f(x) = b^x, where b is a +ve constant not equal to 1, can be written in the alternative form
f(x) = e^kx 
where k is a non-zero constant, given by k = lnb 

Graphs of exponential functions
The graph of the function f(x) = e^kx, where k doesn't equal 0, has the following features 
- the graph lies entirely above the x-axis
- if k > 0, then the graph is increasing, and it gets steeper as x increases
- if k < 0, then the graph is decreasing, and it gets less steep as x increases
- the x-axis is an asymptote
- the y-intercept is 1 
- the closer the value of k is to 0, the flatter the graph 

- the graph of every exponential function is a horizontal scaling of the graph of the exponential function f(x) = e^x 

A characteristic property of exponential growth and decay functions
If f(x) = ae^kx, then whenever p units are added to the value of x, the value of f(x) is multiplied by e^kp (also known as the growth (greater than 1) or decay (between 0 and 1) factor) 

Doubling and halving periods
Suppose that f is an exponential growth function. Then p is the doubling period of f if whenever you add p to x, the value of f(x) doubles.
If f is an exponential decay function, then p is the halving period of f if whenever you add p to x, the value of f(x) halves.

Strategy
To find a doubling or halving period
If f(x) = ae^kx is an exponential growth function (k>0), then the doubling period of f is the solution p of the equation e^kp = 2 or p = (ln2)/k 
If it is an exponential decay function, then the halving period of f is the solution p of the equation e^kp = 1/2 or p (ln(1/2))/k = -(ln2)/k 

Rearranging inequalities
Carrying out any of the following operations on an inequality gives an equivalent inequality
- rearrange the expressions on one or both sides
- swap the sides, provided you reverse the inequality sign
- add or subtract something from both sides
- multiply or divide by something that's positive
- multiply or divide by something that's negative, provided you reverse the inequality sign