**Functions Explained in 5 Minutes**

Functions are a very important concept in mathematics. Here's the basic idea..

Suppose I take a number x and do something with it to produce a new number. That's a function. Mathematicians denote it as f(x), meaning "a function of x".

Think of f as a machine. I feed it the number x, it processes x and spits out a new number. An example will help..

f(x)=3*x+1

Feed in the number x=2 and you get the result f(2)=7. Almost all functions are defined by giving a formula for what they produce, just like the example above.

Try this one..

f(x)=x^2+x+1

So for example f(3)=3^2+3+1=13

Functions are important in many branches of mathematics including calculus. You need to understand the concept of a function before you can understand calculus. That's because much of calculus is about the detailed properties of functions.

Of course, not all functions need to be defined by formulas. Consider this function..

f(x)=x rounded up to the next highest integer. So for example f(3.124)=4

Or try this one, the "sign" function..

f(x)=1 if x is positive

f(x)=-1 if x is negative

But, for much of mathematics a function is given by a formula, such as..

f(x)=10*x+3

f(x)=x^5+x^4+3*x+1

Mathematicians love formulas. I'm not sure why. I suppose a formula makes them feel comfortable.

We can regard functions as just another type of mathematical object and it's even possible to define a "multiply" operation for functions. If f(x) is a function and g(x) is another function we can form a product function f*g defined like this..

First apply the function g to x, then apply the function f to the result. This is written as f(g(x)) and is often abbreviated to fg(x) or f*g or even just fg. But notice that, unlike numbers, fg is not equal to gf. The result depends on the order of multiplication!

Here's an example:

Suppose g(x)=3*x+1 and f(x)=x+10, then we have..

fg=f(g(x))=f(3*x+1)=(3*x+1)+10=3*x+11

but..

gf=g(f(x))=g(x+10)=3*(x+10)+1=3*x+30+1=3*x+31

There's a special case of this. Suppose we have a function g(x) and we manage to find another function f(x) such that fg=f(g(x))=x

This means the product function fg leaves x unchanged. No matter what value of x you feed into fg it will always give you the same exact value back! In this case f is said to be the "inverse" function of g. It's written as g but with a small -1 superscript in the upper right. Not all functions have an inverse, but some do.

For example, the inverse of the function g(x)=x^2 is f(x)=sqrt(x) because fg=f(g(x))=sqrt(x^2)=x

You can also think of an inverse function like this.. no matter what g does to x the inverse function manages to "undo it" and leave x unchanged.

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Content written and posted by Ken Abbott abbottsystems@gmail.com