It's probably a good time to define exactly what a function is. But before we do, let's do a little recap of what we've seen: [br][br]Functions can be simple one step programs like[br][br][code]dianna(x)=3x[/code][br][br]Or functions can have two steps in their code like in the last activity:[br][br][code]f(x)=2*x-1[br][/code][br]Or functions can be complex like the model of the height of a missile from the start of this chapter:[br][code][br]g(x)=-4.92507x^2+297.84768x+61605959[/code][br][br]This is hardly the most complex function either; functions can have as many steps in their code as you want. All of these examples however are really just [i]mathematical computer programs [/i]of varying complexity. Indeed, this is the defining characteristic of a function:[br][br][b]Definition:[/b] A [b]function[/b] is a computer program (or you might call it a "rule" if you prefer) that[br][list][*]takes an input (usually x, but any symbol can be used), and with the input[/*][*]calculates an output using a systematic repeatable process usually involving algebraic manipulation[/*][/list][br]The second bullet point is subtle and yet also critical: the process for calculating the output based on the input [i]must be systematic and repeatable[/i]. So for instance, you can't put 7 into a function in the morning and get 9, and then put 7 in later that day and get 12. This caveat won't flare up much in this book, but if you keep studying mathematics this becomes a big deal.[br][br]We've seen three examples of functions so far. In each of them, the systematic repeatable process was written in algebraic "code" on the right side of an equation. This is normal, but by no means essential. Here's a perfectly good function: "t[code]he function ceiling takes any number as an input and rounds it up to the next largest whole number[/code]". So [code]ceiling(7.2)=8[/code] and [code]ceiling(-7.2)=-7[/code]. There isn't really a great way to use algebra to express [code]ceiling[/code]. The plain English description is the best way to describe it. [br][br]Some functions can be expressed both ways, but are best thought of by their plain English descriptions. For instance consider this one: "[code]the function c takes any number as an input and returns 7 as an output[/code]". So [code]c(5)=7[/code] and [code]c(7)=7[/code]. The function [code]c[/code] is an example of a [b]constant function[/b], because the output never changes. No matter what the input is, the output is always the same, in this case, 7. The algebraic code is [code]c(x)=7[/code] indicating that 7 is the output no matter what the input is. Even though this is such a simple constant function, sometimes people get confused by the algebra code more than the plain English description. So it's good to be familiar with both. [br][br]Constant functions might seem kind of silly, but we'll see they're quite important throughout Calculus because they [i]don't change[/i]. [br][br]Speaking of "change", that reminds me. It's time I tell you why you need to bother knowing about functions. The reason is: [i]the primary objective of Calculus is to study the way functions change[/i]. [br][br]We already saw in [url=https://www.geogebra.org/m/x39ys4d7#material/g7tsaupz]the previous activity[/url] that with code like [code](2+0.1,f(2+0.1))[/code] the function [code]f(x)[/code] didn't change a whole lot when the input was "nudged" by a small amount such as 0.1. This idea of "nudging" the input a little bit with a small number like 0.1 is a core idea in calculus.[br][br]We'll be more specific about all of this later, but--believe or not--you have already done a calculus exercise when you[url=https://www.geogebra.org/m/x39ys4d7#material/j5s3ey98] tried to find the maximum height of the missile[/url]. You might not have noticed it, but in hunting for the maximum, what you were really doing was trying to find the point where the [i]rate of change[/i] of the missile (as modeled by [code]g(x)[/code]) stopped growing (or "changing") for a moment, before it began to decrease. We'll study this in much more detail later.[br][br]When you're ready, move forward to the next activity where we'll let GeoGebra build another function for us: a mathematical model of the length of the day (how long the sun is above the horizon) at [url=https://www.northernvermont.edu/]Northern Vermont University - Johnson[/url].