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Section 1.1 The language of functions and modeling

Welcome to our first video for MATH 3! Today we are going to talk about functions and their properties. You have likely seen this material in one or more of the courses you took before coming to Dartmouth - reviewing it gives us the chance to set up some common vocabulary and notation that we will use throughout the course.

To begin, we think of a function as a connection between a set of inputs and a set of outputs. It is sometimes useful to use the metaphor of a machine - a function is a machine that takes an input, operate on it, and produces an output. Inputs can be anything - numbers, letters, fruits, dogs and cats - as can outputs. In calculus, both inputs and outputs are usually numbers but they can be different sets of numbers. An important restriction on functions is that every input is mapped to a single output, but the same output can occur as the result for more than one input. Let's think about a couple of examples.

In algebra, we manipulate algebraic expressions like \(x^2+2x-1\text{.}\) We might factor it to find its roots, or manipulate it in some other way. But we can also think of it as a function - it is a rule that tells us how operate on any number that we input. If we take the number \(3\text{,}\) we insert it as the \(x\) in the algebraic expression and simplify:

\begin{equation*} (3)^2+2\cdot 3-1 = 9+6-1=14\text{.} \end{equation*}

This function can take any real number as an input and will give a real number as an output.

We often label a function in by writing down such a rule. If we want to call our function \(f\) we would write

\begin{equation*} f(x)=x^2+2x-1. \end{equation*}

This is a short-hand to help us name and identify the process we just executed.

Another example comes from algebra but is a little more complex - the square root function, which we can define as \(g(x)=\sqrt{x}\text{.}\) As with the previous example, we can use this rule to take a number as input and report an output. But, not all numbers make sense as inputs - if the \(x\) we choose is negative, the square root doesn't make sense. So, the set of inputs in the non-negative real numbers. Similarly, the set of outputs is limited as well - the square root function we've defined gives only positive or zero outputs. These two sets - the set of inputs and the set of outputs are called the domain and range of the function respectively. They are important sets to identify when first analyzing a function. We use set notation to describe these sets. For \(g(x)\) we have:

\begin{equation*} D=\{x|x\ge 0\}, \;\; R=\{x|x\ge 0\} \end{equation*}

Here's a quick exercise to test your understanding of the concept of the domain.

One of the most useful things we can do with a function is to plot its graph. This can help us identify the domain and range as well as many other features of the function. Graphing is sometimes difficult - we will spent several classes later in the term thinking about techniques for graphing functions. Right now, we will start with a very straight-forward approach - building a table of inputs and outputs and plotting points.

To do this, we need to select a few points in the domain and then compute the corresponding outputs. In the table below, we show the results of doing this for the inputs of \(\{-3,-2,-1,0,1,2,3\}\)

Table 1.1.1.
x f(x)
-3 2
-2 -1
-1 -2
0 -1
1 2
2 7
3 14
Next, we can plot these on a two-dimensional axis. We will use the horizontal axis for inputs and the vertical axis for output. Figure 1.1.2 shows the results and gives us a bit of a picture of what the function looks like.

Figure 1.1.2. Plotting the outputs of \(f(x)=x^2+2x-2\) for inputs \(\{-3,-2,-1,0,1,2,3\}\) and then a candidate graph for the whole function on this portion of the domain.

If we imagine a curve that passes through these points, the simplest one looks like it decreases, reaches a minimum at \((-1,-2)\) and then increases. But there are lots of other possibilities too. And, there are parts of the domain our plot doesn't cover - like left of \(-3\) and right of \(3\text{.}\) While we can "connect the dots" to get a sense of what the full graph looks like, we can never be sure we have all of the features accounted for. To do that would mean evaluating infinitely many points in the domain. But, rough sketchs like this can be helpful. As we mentioned earlier, we will return to graphing techniques later in the course. For what we do before then, we will often rely on graphs created through computation using a software package.

In our examples above, we used algebraic expressions to define our example functions. This will be the way that we define many of our examples in Calculus. But functions also arise as models. When we look at phenomena in the world, we can try to understand it by collecting data and building theories. For example, the dots in Figure 1.1.2 could be the result of data collection about the height of a flying bird at different points on the ground. We can use functions to create a rule that might describe the observed information - a model of the observed process. This is a very powerful way of tackling "real world" problems and is the foundation of how scientists build theories about the world, make predictions, and then test them.