C3 Functions Round-Up

Reminder: this is really about AQA C3, as OCR and Edexcel are slightly different. They don't change the truths of mathematics though.

After quite a long break, I've started doing maths again. I've forgotten chunks of it, as you would expect. Here are some things I've forgotten (on the assumption that they're generally forgettable):

1. Inverse functions are simply functions reversed. The domain becomes the range and vice versa. However, not all functions can have an inverse. If a function is many-one (as quadratic functions generally are) then they can't have an inverse - it would be one-many, which is not a function at all.

2. To get an inverse function you can do two things. If x appears only once in the original function, then you can do a reverse flow chart, reversing the order and the operations. BUT BEWARE OF SELF INVERSE FUNCTIONS - if you meet a subtraction from or a divide into, those operations stay the same because they're self-inverse functions. IF x appears more than once you need to do some equation-fiddling. On a function, x is a/the domain value(s). y is the same as f(x) - ie it's the range of the function. So by re-arranging the equation to isolate x, you're turning the function INSIDE OUT and making x the f(x) instead of the variable at the beginning (the domain).

3. composite functions are two functions put together. They're usually written as fg(x). What this means is you do both operations. You write out g(x) first, say it's x² + 4. Then you put it into f(x), which might be 2x - 3. The resulting composite function is 2(x²+4)-3.

4. On some functions, there are values that cannot be defined and that affects the range as well as the domain. Take f(x) = 1/x for example. X cannot be 0. But this means there are going to be impossible values of f(x) as well. A good way of estimating this is to draw the graph of f(x) on your calculator!! Always use the calculator wherever possible, even if you need to sketch the graph by more formal methods.

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.

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.