Recall that a function is a rule that takes an input, does something to it,
and gives an output.
Each input has exactly one output.
If the function name is
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$\,f\,$, and the input name is $\,x\,$,
then the unique corresponding output is called $\,f(x)\,$
(which is read aloud as ‘$\,f\,$ of $\,x\,$’).
This use of the notation [beautiful math coming... please be patient] $\,f(x)\,$ to represent the unique output from the function $\,f\,$ when the input is $\,x\,$ is called function notation.
When you're working with a function,
it's critical that you understand the relationship between its inputs and their corresponding outputs.
That is, it's critical that you understand the function's (input,output) pairs.
Of course, there are usually infinitely many of these (input,output) pairs.
For example, consider the squaring function—the function that takes a real number input, and squares it.
When the input is $\,3\,$, the output is $\,3^2 = 9\,$.
Thus,
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$\,(3,9)\,$ is an (input,output) pair.
When the input is $\,4\,$, the output is $\,4^2 = 16\,$.
Thus,
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$\,(4,16)\,$
is an (input,output) pair.
When the input is $\,3\,$, the output is $\,(3)^2 = 9\,$.
Thus,
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$\,(3,9)\,$ is an (input,output) pair.
Here's a table (at right) that summarizes a few of the infinitelymany (input,output) pairs.
Of course, it's impossible to list them all. When these points are plotted in an [beautiful math coming... please be patient] $\,xy\,$coordinate system (see below), with the inputs along the $\,x\,$axis and the outputs along the $\,y\,$axis, a shape clearly emerges in the coordinate plane. 
SOME (INPUT,OUTPUT) PAIRS FOR THE SQUARING FUNCTION

The picture of all the points of the form [beautiful math coming... please be patient] $\,(x,x^2)\,$ is called the graph of the squaring function.
You can explore this graph
using GeoGebra.
GeoGebra is a free, multiplatform, dynamic mathematics software program that joins geometry, algebra and calculus.
(Dr. Fisher pronounces ‘GeoGebra’ like ‘Algebra’ except with a ‘Geo’ at the beginning.)
Click on the link below and have fun! (Please be patient. It may take a few minutes for GeoGebra to load. The link opens in a new window.)
Now it's time to make things precise:
The sketch at right illustrates all the key features of a graph. The input (horizontal) axis is labeled as $\,x\,$. The output (vertical) axis is labeled as $\,y\,$. The graph itself is labeled as $\,y = f(x)\,$. A couple specific (input,output) pairs are shown. You should recognize this as the graph of the squaring function. That is, [beautiful math coming... please be patient] $\,f(x) = x^2\,$. Thus, $\,f(2) = (2)^2 = 4\,$ and $\,f(1) = 1^2 = 1\,$. 
Alternate names for inputs and outputs have been chosen for the graph at left. The input (horizontal) axis is labeled as $\,t\,$. The output (vertical) axis is labeled as $\,w\,$. The graph itself is labeled as $\,w=g(t)\,$. This says that a function named $\,g\,$ is acting on inputs named $\,t\,$ and producing outputs named $\,w\,$. A couple specific (input,output) pairs are shown. You may have guessed that this is the graph of the cubing function. That is, [beautiful math coming... please be patient] $\,g(t) = t^3\,$. Thus, $\,g(1) = (1)^3 = 1\,$ and $\,g(2) = 2^3 = 8\,$. 
You can use
GeoGebra
to play with graphs of functions to your heart's content. Have fun!
(Please be patient. It may take a few minutes for GeoGebra to load.)
Here are two things you will frequently be asked to do:
The graph of a function $\,f\,$ is shown at right. Read the following information from the graph:
