Monday 30 November 2009

The Self-Inverse Function

This sounds really weird or complex but it's not.

Basically a self-inverse function is just a function that gives you the same answer when you do the function to the answer.

Wot?


Alright. Say you have a function f(x) = 4-x

Say x = 2

4-2 = 2.

Now apply the answer to the function

4-2 = 2.

Same answer.


Try again f(x) - 7-x

Say x - 5.

7-5=2.

7-2=5

We have x again.


And f(x) - 10-x

x=4

10-4 = 6

Put 6 into it:

10-6=4.

Brings us back to our starting point with x.


When you do the operation twice, finding the function of the answer, you get your starting value of x.
Reciprocals always give self-inverses too.

When Can an Inverse Function Exist?

You can't always have an inverse function (See previous post or previous but one or thereabouts).

A function is defined as any mapping which is one-one or many-one. This means that for any input value a unique output value is generated.

Something like y= x 2 is a many-one function. 22 and -22 give the same value.

You can always draw a graph of a function.

A function CAN'T be a one-many mapping. It doesn't make a lot of sense to most of us to have some kind of operation which could generate loads of different answers for exactly the same input.

So.

If a function would generate a one-many correspondence, it follows that it's not a function. So a many-one function CANNOT have an inverse, because its inverse would be one-many (ie not a function at all).

Only one-one type functions have inverse functions.

Thursday 19 November 2009

C1 Advice

Hmmm.

I was bored so I thought I'd offer a little advice for anyone revising for C1. As it is a non-calculator paper, you need to consider a few things.

1) Your mental arithmetic should be good
2) You should know square and cube numbers up to and including 53
3) You should be totally au fait handling surds. This is an early part of C1 and you might have forgotten it by the time of the exam. There are some fiddly rules surrounding eliminating surds from equations so learn 'em.
4) You should remember to x stuff by -1 to get rid of unwanted negative numbers in your answers.
5) You need to be good at fractions, including cancelling down.
6) You also should be confident expanding brackets (and other GCSE stuff). This might be annoying if you did GCSE some years ago, like myself. It's worth buying a GCSE revision textbook and looking the Higher level stuff up.
7) Because of the non-calculator thing, the answers to C1 questions are generally nice, like 3 and 5 and 2. If you work something out not in surd form and it is a bizarre fraction a good rule of thumb is to go back and check your working.

Monday 2 November 2009

Functions II - The Inverse

You can reverse a function - not always, to give you what you started with.

This is known as the inverse function

Take y=x2

This function, or y=f(x), will give 4 for the input of 2.

So what will give us 2, from an input of 4?

It would be the opposite, the inverse - x = √y

This is sometimes written as f-1.

It does get more complicated but that is the idiot's version.

A Recommendation

Normally I would not advise students of maths under degree level to go anywhere near Wikipedia. It's not the reliability issue as such, it's that it gets very complicated very quickly.

An exception is the page on Trig. It gives some useful basics and some really handy animations which illustrate the key trigonmetric functions.

Sunday 1 November 2009

Functions

A simple definition of a function is an operation which, for every value put in, will give a single value. It can give the same answer for different inputs. y=x2 will give the answer 4 when both 2 and -2 are put in, but is still a function.

What a function can't do is give multiple answers to the same inputted value. A graph that appears to show this is not the graph of a function.