Understanding Functions

An Inter-DimensionalJunk Food Adventure

This is me,

And this is a function

And this is an Inter-Dimensional  Vending Machine

It will all make sense before the end, I promise.

And now a question: What aremathematical functions?

The Gist of it: functions express relationships

Lets say we have an equation, like this one:

This equation expresses a relationship between x and y  

It says that if we are given the value of x, then we can determine the value of y. For example, if x = 2, then y = 9.

Being able to determine y using only x is what it means to say that y is a function of x.

Writing it down: function notation

Recalling our previous equation

If we want to express that y is a function of x then we  should use function notation.

we can write y = f(x) to mean "y is a function of x".  We often cut out the middle man, y, and just write this 

But there's more: functions are predictableWe can think of a function as a kind of machine that produces an output when it is given an input.                             

Why not?  Well, if x is anything besides zero, then  y is ambiguous. E.g. inputing x=3, outputs either 3 or -3 for y.  

For example, this is an expression in which y is not a function of x :

This machine is predictable, and does not give two different outputs for the same input.  Instead, one input leads to one output.  This is also true of functions.

input output

Some Terminology: domain and range

We have been using the terms input and output instead of the usual terms domain and range.  

The  domain of a function f(x) is anything we can put in for x, and the range of the function is the collection of all of the outputs.

What about that vending machine?

Suppose that we found a vending machine.  Not just any vending machine, but a vending machine that is situated in a multi-dimensional nexus. It never runs out of snacks.

As it turns out, we can think of this inter-dimensional marvel as a function.

What is its domain?

Like most vending machines, you input button presses for the snack you'd like.   

If the vending machine is a function, then ...

What is its range?

When you input some button presses, you get a snack in return.

Is it predictable?

It had better be!  The button for a Choco-Explosion should spit out a Choco-Explosion every time, and nothing else.

For Example

The domain is the set of buttons

The range is the set of snacks

pressing buttons corresponds to plugging in values for x.

A broken machine: recognizing a non-function

Suppose that there is a glitch in the machine.  

We press C3 and hope to get a Mega-Choco Wedge like before.  But this time, a Plutonian Sugar Spin comes out instead.  

This is a clear sign that the machine is dysfunctional, because outputs should be determined by inputs.

Other representations: the many faces of functions

Here is a function in function notation:

And here is a table containing some values of the function for different values of x:

And here is a graph for this function, where y=f(x)

x f(x)1 4-2 133 18

One more thing: evaluating a function

Given a function such as

Then we can evaluate the function by "plugging in" values for x.  For example:

So, evaluating f(x) for x=7, gives 22.

Wrapping Up

Key points:

•  y = f(x) says that the y is a function of x - when we are given x, we can find y.

• The written form f(x) is called function notation.• A function is predictable - two different outputs cannot

be associated with any single input.• Functions can be evaluated by plugging in values for x. • Functions have domain and range, which loosely

correspond to sets of input and output.