Math, The Way it Should Be (Unit 2.3: Functions)

As per usual, this is the latest (and late, sorry) installment of the Math, The Way it Should Be series.  The first post explaining what the point is exists here.  The first unit is here and the beginning of this unit is here.  This unit is all about functions which are pretty cool objects.  Basically, they're boxes that you can put something into and it gives you something else.  Functions are important because if you put the same thing into it twice, you'll always get exactly the same thing back out.  We say that a function maps a domain to a range:

The domain is all of the things that can go into the function and the range is all of the possible things that can come out.  They don't necessarily have to be the same size.  We'll come back to this in a second.  But first, how do we write a function?  Usually it looks a little like this:

\(f(x)\)

The function's name is "f of x" though the x is just a place-holder, really.  To fully define a function, we say what it is and then what its domain is, though this is sometimes omitted when the domain is all numbers.  For example, we could define f this way:

\(f(x)=x^2\)

\( -2 {\leq} x {\leq} 2 \)

This means that whatever goes into f gets squared and only numbers between -2 and 2 may be put into f.  To find out what happens when we put 1 into f, we just write \(f(1)\) and then carry out the function on 1.  We then have:

\(f(1)=1\)

So we know what the function is and it's domain.  What about it's range?  The best way to figure this out is to graph the function.  How do you do that?  Recall the lesson where I introduced the Cartesian plane and said that anything could be put on either axis?  In this case, you put the domain on one axis and the function on the other, like this:

The red line represents the function values.  To find \(f(x)\), just find x on the horizontal axis and then draw a line up to the red line and then over to the vertical axis.  You then have your value.  To make a graph like this, you just come up with a couple examples and write them into pairs in the form \((x, f(x))\) and then put them on the plane.  If you don't remember how to do this, please see the lesson in the link above.  Finally, draw a line through the points and you have your function.  We can now find out what the range of the function is.  The range is the collection of all of the possible outcomes of the function.  In this case, you can see that the lowest value is 0 and the highest value is 4.  So the range is \(0 {\leq} x {\leq} 4\).  And that's all you need to know about functions (for right now)!

Homework:

• Practice with functions.  Start thinking of things in terms of functions and then think about graphing them.  For example, position as a function of time (\(x(t)\)).  Think about what the domain and range might be.  Only one assignment for this lesson!

-Lane