The Modulus Function

Put very basically, the modulus function finds the absolute value of a number.

It is indicated by a pair of straight brackets around an x or a part of an equation.

It just means that -x when operated on by the modulus function becomes x while x stays as x.

It ignores negative values of x, or of the statement in x, such as (x-2).

Transformation of Graphs, contd

When you are doing composite transformation of graphs, as you will at C3 and presumably C4 level, you do need to remember that sometimes the order of transformation counts. I have to admit to struggling with this (I don't have a tutor right now) so forgive me if I fail to offer a convincing explanation. But it's something like this:

If you are stretching in the x-direction, that comes BEFORE any translation.

If you are stretching in the y-direction, that comes AFTER any translation.

That's what I've picked up from getting lots of answers wrong, anyhow. I think it's to do with the way that a y-stretch affects the whole equation, while an x-stretch affects anything directly attached to the x part of the equation.

Some websites do suggest that you start at the innermost point of the equation and then work out - I don't know if this works for all composite transformations. I doubt it if what I've written above is right.

More LaTeX Experiments: Transformation of Graphs

If you take a graph and transform it in various ways, here is what you get:

A reflection in the y-axis turns a graph of y=f(x) into the graph of y= f(-x).

A reflection in the x-axis turns a graph of y=f(x) into the graph of y= -f(x)

- note the difference -

A stretch of factor s in the y- direction turns the graph of y= f(x) into the graph of

y = s f(x).

A stretch of factor m in the x-direction turns the graph of y = f(x) into the graph of \[y = f(\frac{x}{m})\]

A translation by \[ \left( \frac{k}{l} \right) \] turns the graph of y = f(x) into the graph of y = f(x-k) + l

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.

e (2.71828)

No, this is not a post for luvved-up early 90s teenagers.

e is a bizarrely cool number, sometimes known as the natural logarithm.

Like π it is an irrational number - ie it cannot be expressed exactly as the ratio of two whole numbers (and thus as a fraction).

How do we get it then?

Imagine an exponential curve. A curve that is n^x. Say, y= 3^x.

As usual, the gradient will generally differ as x differs.

Is there a curve which has a gradient of 1, where x=0? There is, and, coolly, this curve has a gradient of 1 where x=0 and ALSO passes through y=1, ie the value of the function is 1.

This curve is y = e ^x. It's not just at y=1 where the value of the function equals the gradient, it's every point on the curve.

As you can imagine, that makes differentiating with e easy...

There is an excellent page here which shows you some interesting examples of e in action.

Reciprocal Trig Functions

Sine, cosine and tangent are functions. You input a value - an angle in radians, or degrees, and for each angle there is a value. They are repeating functions.

They also have reciprocal functions.

The reciprocal of sine x is cosec x and is (1/sin x)

The reciprocal of cosine x is secant x and is (1/cos x)

The reciprocal of tan x is cotangent x (cot x) and is (1/tan x).

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.

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.

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.