Hi John,

I find it helps sometimes to think of a function as a machine, one where you give a number as input to the machine and receive a number as the output. The name of the function is $f,$ the input is $x$ and the output is $f(x),$ read "$f$ of $x".$ The output $f(x)$ is sometimes given an additional name $y$ by $y = f(x).$

The example that comes to mind is the square root function on your calculator. The name of the function is $\sqrt{\;\;}$ and we usually write the function as $f(x) = \sqrt{x}.$ On my calculator I input $x$ for example by pressing 2 then 5. Then I invoke the function by pressing the $\sqrt{\;\;}$ button I and receive 5 on the calculator display. Thus $\sqrt{25} = 5.$

Let me look at another example, $y = f(x) = x^2 - 4.$ In this example I am describing the function $f$ by the way it operates. By $f(x) = x^2 - 4$ I am telling you that if you input a number $x$ to this function then the function squares $x,$ subtracts 4 and returns the result. Thus for example if $x = 3$ then $y = f(3) = 3^2 - 4 = 9 - 4 = 5.$ To graph this function I would start by choosing some values of $x$ and since I get to choose I would select values that make the arithmetic easy. For example $x = 0, x = 1, x = -1$ and so on. I am going to keep track of what I am doing by using a table.

Add a few more rows to the table choosing your own values of $x.$ Plot the values in your table $(0, -4), (1, -3)$ etc. on graph paper and then see if you can sketch a graph that goes through these points.

Write back if you need more help,
Penny