Trevor Pythagoras Maths » functions

Monotonic (Increasing and Decreasing) Functions

admin — Fri, 22 Jan 2010 16:56:11 +0000

Monotonic functions are functions on real numbers which are either always increasing or always decreasing. Monotonic is a collective name for increasing, strictly increasing, decreasing and strictly decreasing functions.

These can be defined as follows,
if for all x₁ < x₂

f(x₁) ≤ f(x₂) then f is increasing
f(x₁ ) < f(x₁ ) then f is strictly increasing
f(x₁ ) ≥ f(x₂) the f is decreasing
f(x₁) < f(x₂) then f is strictly decreasing

Monotonicity and Derivatives

If a function f(x) is increasing then what we mean is that the slope is always positive, so if f is continuous we can relate the the properties of increasing and decreasing to the derivative as shown in the table below.

Increasing/Decreasing	condition of f’(x)
Increasing	f’(x) ≥ 0
Strictly Increasing	f’(x) > 0
Decreasing	f’(x)≤ 0
Strictly Decreasing	f’(x) <0

Its important to note that these rules only work if the function is continuous, for example consider f(x) =1/x, which is discontinuous at 0.
We can differentiate it to get f’(x) = -1/x² which we know is always negative (because the squared term is always positive) so we would expect it to be a decreasing function. However if we consider two point either side o, 1 and -1 say we find that f is not a decreasing function because whilst -1 < 1, -1/x < 1/x contrary to our definition of decreasing

Also see

Finding the Inverse of a Function

admin — Wed, 20 Jan 2010 18:01:30 +0000

Before you can try and find the inverse of a function you need to determine if one exists. For an inverse to exist exactly one element of the domain must map to each element of the co-domain (though you can always re-define the co-domain to only include elements that are mapped to).

So for example f(x) = x+2 (from reals to reals) has an inverse as every element y is mapped to only by y-2 but f(x) = x² (from reals to reals) doesnt because both -1 and 1 map to 1 so how would we decide which of these would be the unique result of the inverse applied to 1 (we could of course define the domain to be the positive reals in which case it would have an inverse).

When finding the inverse of a function you are really looking to see what maps to each element of the range or codomain, to find the inverse of f you are looking for the element x in the domain for a given y in the range such that f(x) = y. This basically means you are reversing the process of the function.

Finding the Inverse when the function is a formula

When a function is given by a formula what you need to try and do is apply the operations of that formula backwards. This easiest way of doing this is to let the function f(x) = y. Now you know the formula to get from x to y so substitue this in for f(x). All that now needs to be done is to rearrange this equation so that x is the subject and the resulting rexpression on the otherside is only in terms of y. This function is the inverse of f, to show this lets denote it g so we g(y) = x.

Then g(f(x)) = g(y) = x since we started by letting f(x) = y and created g such that g(y) =x.
and f(g(y)) = f(x) = y

so g is indeed the inverse of f.

Example

This is most easily demostrated through an example.
Let
Then to find the inverse of f we first write

and rearrange as follows

So the inverse of f is given by the formula so we can write

Note that we have replaced the y’s with x’s, this doesnt matter as we can put any variable we like into the function but it important to make sure that you use the same variable as the parameter of the function and in the formula that defines it.

Also see

Introduction to functions and maps

admin — Tue, 19 Jan 2010 21:16:48 +0000

A function or map is a way taking elements from one set and using them to find an element of another set. Usually in when you are first learning about functions both of these sets are taken to be the real numbers ( ie 1,2,1/2,pi etc). So any function must have three things:

A domain
A co-domain or range
A “rule” for assigning each element of the domain to a unique element of the co-domain.

Important Points

There are some important things about functions, in particular part 3, which you need to remember.

Firstly the function must be able to be applied to every element of the domain so f(x) = 1/x isn’t a function from the real numbers to the real numbers since 0 cant be assigned a value, as 1/0 isn’t a real number. There are two ways round this problem, we can define the co-domain not to include 0 (ie ) or can give 1/0 a different rule, eg f(x) = 1/x for all and f(0) = 7.

However not every element in the co-domain needs to be assigned to something so is a valid function even though nothing goes to -1

Every element of the domain is assigned a unique element of the c0-domain. This uniqueness is often a source of confusion – here what we mean is that the function only assigns a single element of the co-domain to each element of the domain, but more than one element of the domain can be assigned the same element of the co-domain.

Simple examples

Here are some simple examples of functions from the real numbers to the real numbers.

– This simply applys this operation to any number and give you another number.

for $ x \not 0 $ and – This takes the reciprocal of a number unless that number is zero where the reciprocal is undefined so gives 7.

-Note that f(a) = f(-a) and nothing gives f(x) = -1 however this is still a function as f(x) is defined for all x and f(x) gives only one answer.

Composing Functions

You can compose two or more functions to form a new function. Consider two functions f and g both from the reals to the reals, f composed with g, written, fg or is given by

so if f = x+1 and g=2x fg(x) = f(2x) = 2x+1

Note that in order to do this the domain of g has to be the same as the co-domain of f.

Inverses and the Identity map

The identity map or function is the function which does nothing. If I is the identity on the reals the I(x) = x.

The inverse of a function , written , is the function which when composed with f gives the identity. There are two types of inverse, a right sided inverse and a left sided inverse, depending upon which side you compose them with f.

g is the left sided inverse of f is

gf(x) = I(x) = x

and h is the right sided inverse of f if

fh(x) = I(x) = x

If there is a function h such that

hf = fh = I

then h is said to be a two sided inverse of f and is often simply refereed to as the inverse of f.

Note that not all functions have inverses, for example x² has no inverse, but when a function does have one it is unique.